Preprint typeset in JHEP style - HYPER VERSION

The Multiverse Interpretation of Quantum

Mechanics

Raphael Boussoab and Leonard Susskindc

a Center for Theoretical Physics Department of Physics

University of California Berkeley CA 94720 USAb Lawrence Berkeley National Laboratory Berkeley CA 94720 USAc Stanford Institute for Theoretical Physics and

Department of Physics Stanford University Stanford CA 94305 USA

Abstract We argue that the many-worlds of quantum mechanics and the many

worlds of the multiverse are the same thing and that the multiverse is necessary to

give exact operational meaning to probabilistic predictions from quantum mechanics

Decoherencemdashthe modern version of wave-function collapsemdashis subjective in that it

depends on the choice of a set of unmonitored degrees of freedom the ldquoenvironmentrdquo

In fact decoherence is absent in the complete description of any region larger than

the future light-cone of a measurement event However if one restricts to the causal

diamondmdashthe largest region that can be causally probedmdashthen the boundary of the

diamond acts as a one-way membrane and thus provides a preferred choice of environ-

ment We argue that the global multiverse is a representation of the many-worlds (all

possible decoherent causal diamond histories) in a single geometry

We propose that it must be possible in principle to verify quantum-mechanical pre-

dictions exactly This requires not only the existence of exact observables but two

additional postulates a single observer within the universe can access infinitely many

identical experiments and the outcome of each experiment must be completely definite

In causal diamonds with finite surface area holographic entropy bounds imply that no

exact observables exist and both postulates fail experiments cannot be repeated in-

finitely many times and decoherence is not completely irreversible so outcomes are

not definite We argue that our postulates can be satisfied in ldquohatsrdquo (supersymmetric

multiverse regions with vanishing cosmological constant) We propose a complemen-

tarity principle that relates the approximate observables associated with finite causal

diamonds to exact observables in the hat

arX

iv1

105

3796

v3 [

hep-

th]

22

Jul 2

011

Contents

1 Introduction 1

2 Building the multiverse from the many worlds of causal diamonds 6

21 Decoherence and causality 6

22 Failure to decohere A problem with the global multiverse 8

23 Simpliciorsquos proposal 10

24 Objective decoherence from the causal diamond 12

25 Global-local measure duality 17

26 Constructing a global multiverse from many causal diamond worlds 20

3 The many worlds of the census taker 24

31 Decoherence and recoherence 24

32 Failure to irreversibly decohere A limitation of finite systems 26

33 Sagredorsquos postulates 27

34 Irreversible decoherence and infinite repetition in the hat 33

35 Black hole complementarity and hat complementarity 36

36 The global multiverse in a hat 41

1 Introduction

According to an older view of quantum mechanics objective phenomena only occur

when an observation is made and as a result the wave function collapses A more

modern view called decoherence considers the effects of an inaccessible environment

that becomes entangled with the system of interest (including the observer) But at

what point precisely do the virtual realities described by a quantum mechanical wave

function turn into objective realities

This question is not about philosophy Without a precise form of decoherence

one cannot claim that anything really ldquohappenedrdquo including the specific outcomes of

experiments And without the ability to causally access an infinite number of pre-

cisely decohered outcomes one cannot reliably verify the probabilistic predictions of a

quantum-mechanical theory

ndash 1 ndash

The purpose of this paper is to argue that these questions may be resolved by

cosmology We will offer some principles that we believe are necessary for a consistent

interpretation of quantum mechanics and we will argue that eternal inflation is the

only cosmology which satisfies those principles There are two views of an eternally

inflating multiverse global (or parallel) vs local (or series) The parallel view is like

looking at a tree and seeing all its branches and twigs simultaneously The series view

is what is seen by an insect climbing from the base of the tree to a particular twig along

a specific route

In both the many-worlds interpretation of quantum mechanics and the multiverse

of eternal inflation the world is viewed as an unbounded collection of parallel universes

A view that has been expressed in the past by both of us is that there is no need to

add an additional layer of parallelism to the multiverse in order to interpret quantum

mechanics To put it succinctly the many-worlds and the multiverse are the same

thing [1]

Decoherence Decoherence1 explains why observers do not experience superpositions

of macroscopically distinct quantum states such as a superposition of an alive and a

dead cat The key insight is that macroscopic objects tend to quickly become entangled

with a large number of ldquoenvironmentalrdquo degrees of freedom E such as thermal photons

In practice these degrees of freedom cannot be monitored by the observer Whenever

a subsystem E is not monitored all expectation values behave as if the remaining

system is in a density matrix obtained by a partial trace over the Hilbert space of E

The density matrix will be diagonal in a preferred basis determined by the nature of

the interaction with the environment

As an example consider an isolated quantum system S with a two-dimensional

Hilbert space in the general state a|0〉S + b|1〉S Suppose a measurement takes place in

a small spacetime region which we may idealize as an event M By this we mean that

at M the system S interacts and becomes correlated with the pointer of an apparatus

A

(a|0〉S + b|1〉S)otimes |0〉A rarr a |0〉S otimes |0〉A + b |1〉S otimes |1〉A (11)

This process is unitary and is referred to as a pre-measurement

We assume that the apparatus is not a closed system (This is certainly the case

in practice for a macroscopic apparatus) Thus shortly afterwards (still at M in our

idealization) environmental degrees of freedom E scatter off of the apparatus and

become entangled with it By unitarity the system SAE as a whole remains in a pure

1For reviews see [2 3] For a pedagogical introduction see [4]

ndash 2 ndash

quantum state2

|ψ〉 = a |0〉S otimes |0〉A otimes |0〉E + b |1〉S otimes |1〉A otimes |1〉E (12)

We assume that the observer does not monitor the environment therefore he will

describe the state of SA by a density matrix obtained by a partial trace over the

Hilbert space factor representing the environment

ρSA = TrE|ψ〉〈ψ| (13)

This matrix is diagonal in the basis |0〉S otimes |0〉A |0〉S otimes |1〉A |1〉S otimes |0〉A |1〉S otimes |1〉A of

the Hilbert space of SA

ρSA = diag(|a|2 0 0 |b|2) (14)

This corresponds to a classical ensemble in which the pure state |0〉S otimes |0〉A has prob-

ability |a|2 and the state |1〉S otimes |1〉A has probability |b|2

Decoherence explains the ldquocollapse of the wave functionrdquo of the Copenhagen in-

terpretation as the non-unitary evolution from a pure to a mixed state resulting from

ignorance about an entangled subsystem E It also explains the very special quantum

states of macroscopic objects we experience as the elements of the basis in which the

density matrix ρSA is diagonal This preferred basis is picked out by the apparatus

configurations that scatter the environment into orthogonal states Because interac-

tions are usually local in space ρSA will be diagonal with respect to a basis consisting

of approximate position space eigenstates This explains why we perceive apparatus

states |0〉A (pointer up) or |1〉A (pointer down) but never the equally valid basis states

|plusmn〉A equiv 2minus12(|0〉Aplusmn|1〉A) which would correspond to superpositions of different pointer

positions

The entangled state obtained after premeasurement Eq (11) is a superposition

of two unentangled pure states or ldquobranchesrdquo In each branch the observer sees a

definite outcome |0〉 or |1〉 This in itself does not explain however why a definite

outcome is seen with respect to the basis |0〉 |1〉 rather than |+〉 |minus〉 Because

the decomposition of Eq (11) is not unique [5] the interaction with an inaccessible

environment and the resulting density matrix are essential to the selection of a preferred

basis of macroscopic states

Decoherence has two important limitations it is subjective and it is in principle

reversible This is a problem if we rely on decoherence for precise tests of quantum

2We could explicitly include an observer who becomes correlated to the apparatus through inter-

action with the environment resulting in an entangled pure state of the form a|0〉S otimes |0〉A otimes |0〉E otimes|0〉O + b|1〉S otimes |1〉A otimes |1〉E otimes |1〉O For notational simplicity we will subsume the observer into A

ndash 3 ndash

mechanical predictions We argue in Sec 2 that causal diamonds provide a natural

definition of environment in the multiverse leading to an observer-independent notion

of decoherent histories In Sec 3 we argue that these histories have precise irreversible

counterparts in the ldquohatrdquo-regions of the multiverse We now give a more detailed

overview of this paper

Outline In Sec 2 we address the first limitation of decoherence its subjectivity Be-

cause coherence is never lost in the full Hilbert space SAE the speed extent and

possible outcomes of decoherence depend on the definition of the environment E This

choice is made implicitly by an observer based on practical circumstances the envi-

ronment consists of degrees of freedom that have become entangled with the system

and apparatus but remain unobserved It is impractical for example to keep track of

every thermal photon emitted by a table of all of its interactions with light and air

particles and so on But if we did then we would find that the entire system SAE

behaves as a pure state |ψ〉 which may be a ldquocat staterdquo involving the superposition

of macroscopically different matter configurations Decoherence thus arises from the

description of the world by an observer who has access only to a subsystem To the

extent that the environment is defined by what a given observer cannot measure in

practice decoherence is subjective

The subjectivity of decoherence is not a problem as long as we are content to

explain our own experience ie that of an observer immersed in a much larger system

But the lack of any environment implies that decoherence cannot occur in a complete

unitary description of the whole universe It is possible that no such description exists

for our universe In Sec 21 we will argue however that causality places restrictions

on decoherence in much smaller regions in which the applicability of unitary quantum-

mechanical evolution seems beyond doubt

In Sec 22 we apply our analysis of decoherence and causality to eternal infla-

tion We will obtain a straightforward but perhaps surprising consequence in a global

description of an eternally inflating spacetime decoherence cannot occur so it is in-

consistent to imagine that pocket universes or vacuum bubbles nucleate at particular

locations and times In Sec 23 we discuss a number of attempts to rescue a unitary

global description and conclude that they do not succeed

In Sec 24 we review the ldquocausal diamondrdquo description of the multiverse The

causal diamond is the largest spacetime region that can be causally probed and it can

be thought of as the past light-cone from a point on the future conformal boundary

We argue that the causal diamond description leads to a natural observer-independent

choice of environment because its boundary is light-like it acts as a one-way membrane

and degrees of freedom that leave the diamond do not return except in very special

ndash 4 ndash

cases These degrees of freedom can be traced over leading to a branching tree of

causal diamond histories

Next we turn to the question of whether the global picture of the multiverse can

be recovered from the decoherent causal diamonds In Sec 25 we review a known

duality between the causal diamond and a particular foliation of the global geometry

known as light-cone time both give the same probabilities This duality took the

standard global picture as a starting point but in Sec 26 we reinterpret it as a way of

reconstructing the global viewpoint from the local one If the causal diamond histories

are the many-worlds this construction shows that the multiverse is the many-worlds

pieced together in a single geometry

In Sec 3 we turn to the second limitation associated with decoherence its re-

versibility Consider a causal diamond with finite maximal boundary area Amax En-

tropy bounds imply that such diamonds can be described by a Hilbert space with finite

dimension no greater than exp(Amax2) [67]3 This means that no observables in such

diamonds can be defined with infinite precision In Sec 31 and 32 we will discuss

another implication of this finiteness there is a tiny but nonzero probability that deco-

herence will be undone This means that the decoherent histories of causal diamonds

and the reconstruction of a global spacetime from such diamonds is not completely

exact

No matter how good an approximation is it is important to understand the precise

statement that it is an approximation to In Sec 33 we will develop two postulates

that should be satisfied by a fundamental quantum-mechanical theory if decoherence

is to be sharp and the associated probabilities operationally meaningful decoherence

must be irreversible and it must occur infinitely many times for a given experiment in

a single causally connected region

The string landscape contains supersymmetric vacua with exactly vanishing cosmo-

logical constant Causal diamonds which enter such vacua have infinite boundary area

at late times We argue in Sec 34 that in these ldquohatrdquo regions all our postulates can

be satisfied Exact observables can exist and decoherence by the mechanism of Sec 24

can be truly irreversible Moreover because the hat is a spatially open statistically

homogeneous universe anything that happens in the hat will happen infinitely many

times

In Sec 35 we review black hole complementarity and we conjecture an analogous

ldquohat complementarityrdquo for the multiverse It ensures that the approximate observables

and approximate decoherence of causal diamonds with finite area (Sec 24) have precise

counterparts in the hat In Sec 36 we propose a relation between the global multiverse

3This point has long been emphasized by Banks and Fischler [8ndash10]

ndash 5 ndash

M

E S+A

Figure 1 Decoherence and causality At the event M a macroscopic apparatus A becomes

correlated with a quantum system S Thereafter environmental degrees of freedom E interact

with the apparatus In practice an observer viewing the apparatus is ignorant of the exact

state of the environment and so must trace over this Hilbert space factor This results in a

mixed state which is diagonal in a particular ldquopointerrdquo basis picked out by the interaction

between E and A The state of the full system SAE however remains pure In particular

decoherence does not take place and no preferred bases arises in a complete description of

any region larger than the future lightcone of M

reconstruction of Sec 26 and the Census Taker cutoff [11] on the hat geometry

Two interesting papers have recently explored relations between the many-worlds

interpretation and the multiverse [12 13] The present work differs substantially in a

number of aspects Among them is the notion that causal diamonds provide a pre-

ferred environment for decoherence our view of the global multiverse as a patchwork

of decoherent causal diamonds our postulates requiring irreversible entanglement and

infinite repetition and the associated role we ascribe to hat regions of the multiverse

2 Building the multiverse from the many worlds of causal di-

amonds

21 Decoherence and causality

The decoherence mechanism reviewed above relies on ignoring the degrees of freedom

that a given observer fails to monitor which is fine if our goal is to explain the ex-

ndash 6 ndash

periences of that observer But this subjective viewpoint clashes with the impersonal

unitary description of large spacetime regionsmdashthe viewpoint usually adopted in cos-

mology We are free of course to pick any subsystem and trace over it But the

outcome will depend on this choice The usual choices implicitly involve locality but

not in a unique way

For example we might choose S to be an electron and E to be the inanimate

laboratory The systemrsquos wave function collapses when the electron becomes entangled

with some detector But we may also include in S everything out to the edge of the

solar system The environment is whatever is out beyond the orbit of Pluto In that

case the collapse of the system wavefunction cannot take place until a photon from the

detector has passed Plutorsquos orbit This would take about a five hours during which the

system wavefunction is coherent

In particular decoherence cannot occur in the complete quantum description of

any region larger than the future light-cone of the measurement event M (Fig 1) All

environmental degrees of freedom that could have become entangled with the apparatus

since the measurement took place must lie within this lightcone and hence are included

not traced over in a complete description of the state An example of such a region

is the whole universe ie any Cauchy surface to the future of M But at least at

sufficiently early times the future light-cone of M will be much smaller than the whole

universe Already on this scale the system SAE will be coherent

In our earlier example suppose that we measure the spin of an electron that is

initially prepared in a superposition of spin-up and spin-down a|0〉S + b|1〉S resulting

in the state |ψ〉 of Eq (12) A complete description of the solar system (defined as

the interior of a sphere the size of Plutorsquos orbit with a light-crossing time of about

10 hours) by a local quantum field theory contains every particle that could possibly

have interacted with the apparatus after the measurement for about 5 hours This

description would maintain the coherence of the macroscopic superpositions implicit in

the state |ψ〉 such as apparatus-up with apparatus-down until the first photons that

are entangled with the apparatus leave the solar system

Of course a detailed knowledge of the quantum state in such large regions is

unavailable to a realistic observer (Indeed if the region is larger than a cosmological

event horizon then its quantum state is cannot be probed at all without violating

causality) Yet our theoretical description of matter fields in spacetime retains in

principle all degrees of freedom and full coherence of the quantum state In theoretical

cosmology this can lead to inconsistencies if we describe regions that are larger than the

future light-cones of events that we nevertheless treat as decohered We now consider

an important example

ndash 7 ndash

22 Failure to decohere A problem with the global multiverse

The above analysis undermines what we will call the ldquostandard global picturerdquo of an

eternally inflating spacetime Consider an effective potential containing at least one

inflating false vacuum ie a metastable de Sitter vacuum with decay rate much less

than one decay per Hubble volume and Hubble time We will also assume that there

is at least one terminal vacuum with nonpositive cosmological constant (The string

theory landscape is believed to have at least 10100primes of vacua of both types [14ndash17])

According to the standard description of eternal inflation an inflating vacuum nu-

cleates bubble-universes in a statistical manner similar to the way superheated water

nucleates bubbles of steam That process is described by classical stochastic production

of bubbles which occurs randomly but the randomness is classical The bubbles nucle-

ate at definite locations and coherent quantum mechanical interference plays no role

The conventional description of eternal inflation similarly based on classical stochastic

processes However this picture is not consistent with a complete quantum-mechanical

description of a global region of the multiverse

To explain why this is so consider the future domain of dependence D(Σ0) of a

sufficiently large hypersurface Σ0 which need not be a Cauchy surface D(Σ0) consists

of all events that can be predicted from data on Σ0 see Fig 2 If Σ0 contains suffi-

ciently large and long-lived metastable de Sitter regions then bubbles of vacua of lower

energy do not consume the parent de Sitter vacua in which they nucleate [18] Hence

the de Sitter vacua are said to inflate eternally producing an unbounded number of

bubble universes The resulting spacetime is said to have the structure shown in the

conformal diagram in Fig 2 with bubbles nucleating at definite spacetime events The

future conformal boundary is spacelike in regions with negative cosmological constant

corresponding to a local big crunch The boundary contains null ldquohatsrdquo in regions

occupied by vacua with Λ = 0

But this picture does not arise in a complete quantum description of D(Σ0) The

future light-cones of events at late times are much smaller than D(Σ0) In any state

that describes the entire spacetime region D(Σ0) decoherence can only take place at

the margin of D(Σ0) (shown light shaded in Fig 2) in the region from which particles

can escape into the complement of D(Σ0) in the full spacetime No decoherence can

take place in the infinite spacetime region defined by the past domain of dependence

of the future boundary of D(Σ0) In this region quantum evolution remains coherent

even if it results in the superposition of macroscopically distinct matter or spacetime

configurations

An important example is the superposition of vacuum decays taking place at dif-

ferent places Without decoherence it makes no sense to say that bubbles nucleate at

ndash 8 ndash

future boundary

Σ0

Figure 2 The future domain of dependence D(Σ0) (light or dark shaded) is the spacetime

region that can be predicted from data on the timeslice Σ0 If the future conformal boundary

contains spacelike portions as in eternal inflation or inside a black hole then the future

light-cones of events in the dark shaded region remain entirely within D(Σ0) Pure quantum

states do not decohere in this region in a complete description of D(Σ0) This is true even for

states that involve macroscopic superpositions such as the locations of pocket universes in

eternal inflation (dashed lines) calling into question the self-consistency of the global picture

of eternal inflation

particular times and locations rather a wavefunction with initial support only in the

parent vacuum develops into a superposition of parent and daughter vacua Bubbles

nucleating at all places at times are ldquoquantum superimposedrdquo With the gravitational

backreaction included the metric too would remain in a quantum-mechanical super-

position This contradicts the standard global picture of eternal inflation in which

domain walls vacua and the spacetime metric take on definite values as if drawn from

a density matrix obtained by tracing over some degrees of freedom and as if the inter-

action with these degrees of freedom had picked out a preferred basis that eliminates

the quantum superposition of bubbles and vacua

Let us quickly get rid of one red herring Can the standard geometry of eternal

inflation be recovered by using so-called semi-classical gravity in which the metric is

sourced by the expectation value of the energy-momentum tensor

Gmicroν = 8π〈Tmicroν〉 (21)

This does not work because the matter quantum fields would still remain coherent At

the level of the quantum fields the wavefunction initially has support only in the false

vacuum Over time it evolves to a superposition of the false vacuum (with decreasing

amplitude) with the true vacuum (with increasing amplitude) plus a superposition

of expanding and colliding domain walls This state is quite complicated but the

expectation value of its stress tensor should remain spatially homogeneous if it was so

initially The net effect over time would be a continuous conversion of vacuum energy

into ordinary matter or radiation (from the collision of bubbles and motion of the scalar

field) By Eq (21) the geometry spacetime would respond to the homogeneous glide

ndash 9 ndash

of the vacuum energy to negative values This would result in a global crunch after

finite time in stark contrast to the standard picture of global eternal inflation In

any case it seems implausible that semi-classical gravity should apply in a situation in

which coherent branches of the wavefunction have radically different gravitational back-

reaction The AdSCFT correspondence provides an explicit counterexample since the

superposition of two CFT states that correspond to different classical geometries must

correspond to a quantum superposition of the two metrics

The conclusion that we come to from these considerations is not that the global

multiverse is meaningless but that the parallel view should not be implemented by

unitary quantum mechanics But is there an alternative Can the standard global

picture be recovered by considering an observer who has access only to some of the

degrees of freedom of the multiverse and appealing to decoherence We debate this

question in the following section

23 Simpliciorsquos proposal

Simplicio and Sagredo have studied Sections 21 and 22 supplied to them by Salviati

They meet at Sagredorsquos house for a discussion

Simplicio You have convinced me that a complete description of eternal inflation

by unitary quantum evolution on global slices will not lead to a picture in which bubbles

form at definite places and times But all I need is an observer somewhere Then I

can take this observerrsquos point of view and trace over the degrees of freedom that are

inaccessible to him This decoheres events such as bubble nucleations in the entire

global multiverse It actually helps that some regions are causally disconnected from the

observer this makes his environmentmdashthe degrees of freedom he fails to accessmdashreally

huge

Sagredo An interesting idea But you seem to include everything outside the

observerrsquos horizon region in what you call the environment Once you trace over it it

is gone from your description and you could not possibly recover a global spacetime

Simplicio Your objection is valid but it also shows me how to fix my proposal

The observer should only trace over environmental degrees in his own horizon Deco-

herence is very efficient so this should suffice

Sagredo I wonder what would happen if there were two observers in widely

separated regions If one observerrsquos environment is enough to decohere the whole

universe which one should we pick

Simplicio I have not done a calculation but it seems to me that it shouldnrsquot

matter The outcome of an experiment by one of the observers should be the same no

matter which observerrsquos environment I trace over That is certainly how it works when

you and I both stare at the same apparatus

ndash 10 ndash

future boundary

Σ0

P O

Figure 3 Environmental degrees of freedom entangled with an observer at O remain within

the causal future of the causal past of O J+[Jminus(O)] (cyanshaded) They are not entangled

with distant regions of the multiverse Tracing over them will not lead to decoherence of

a bubble nucleated at P for example and hence will fail to reproduce the standard global

picture of eternal inflation

Sagredo Something is different about the multiverse When you and I both

observe Salviati we all become correlated by interactions with a common environment

But how does an observer in one horizon volume become correlated with an object in

another horizon volume far away

Salviati Sagredo you hit the nail on the head Decoherence requires the inter-

action of environmental degrees of freedom with the apparatus and the observer This

entangles them and it leads to a density matrix once the environment is ignored by the

observer But an observer cannot have interacted with degrees of freedom that were

never present in his past light-cone

Sagredo Thank you for articulating so clearly what to me was only a vague

concern Simplicio you look puzzled so let me summarize our objection in my own

words You proposed a method for obtaining the standard global picture of eternal

inflation you claim that we need only identify an arbitrary observer in the multiverse

and trace over his environment If we defined the environment as all degrees of freedom

the observer fails to monitor then it would include the causally disconnected regions

outside his horizon With this definition these regions will disappear entirely from your

description in conflict with the global picture So we agreed to define the environment

as the degrees of freedom that have interacted with the observer and which he cannot

access in practice But in this case the environment includes no degrees of freedom

outside the causal future of the observerrsquos causal past I have drawn this region in

Fig 3 But tracing over an environment can only decohere degrees of freedom that it

is entangled with In this case it can decohere some events that lie in the observerrsquos

past light-cone But it cannot affect quantum coherence in far-away horizon regions

because the environment you have picked is not entangled with these regions In those

ndash 11 ndash

regions bubble walls and vacua will remain in superposition which again conflicts with

the standard global picture of eternal inflation

Simplicio I see that my idea still has some problems I will need to identify more

than one observer-environment pair In fact if I wish to preserve the global picture

of the multiverse I will have to assume that an observer is present in every horizon

volume at all times Otherwise there will be horizon regions where no one is around

to decide which degrees of freedom are hard to keep track of so there is no way to

identify and trace over an environment In such regions bubbles would not form at

particular places and times in conflict with the standard global picture

Sagredo But this assumption is clearly violated in many landscape models Most

de Sitter vacua have large cosmological constant so that a single horizon volume is too

small to contain the large number of degrees of freedom required for an observer And

regions with small vacuum energy may be very long lived so the corresponding bubbles

contain many horizon volumes that are completely empty Irsquom afraid Simplicio that

your efforts to rescue the global multiverse are destined to fail

Salviati Why donrsquot we back up a little and return to Simpliciorsquos initial sug-

gestion Sagredo you objected that everything outside an observerrsquos horizon would

naturally be part of his environment and would be gone from our description if we

trace over it

Sagredo which means that the whole global description would be gone

Salviati but why is that a problem No observer inside the universe can ever

see more than what is in their past light-cone at late times or more precisely in their

causal diamond We may not be able to recover the global picture by tracing over

the region behind an observerrsquos horizon but the same procedure might well achieve

decoherence in the region the observer can actually access In fact we donrsquot even

need an actual observer we can get decoherence by tracing over degrees of freedom

that leave the causal horizon of any worldline This will allow us to say that a bubble

formed in one place and not another So why donrsquot we give up on the global description

for a moment Later on we can check whether a global picture can be recovered in

some way from the decoherent causal diamonds

Salviati hands out Sections 24ndash26

24 Objective decoherence from the causal diamond

If Hawking radiation contains the full information about the quantum state of a star

that collapsed to form a black hole then there is an apparent paradox The star is

located inside the black hole at spacelike separation from the Hawking cloud hence two

copies of the original quantum information are present simultaneously The xeroxing of

quantum information however conflicts with the linearity of quantum mechanics [19]

ndash 12 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

Contents

1 Introduction 1

2 Building the multiverse from the many worlds of causal diamonds 6

21 Decoherence and causality 6

22 Failure to decohere A problem with the global multiverse 8

23 Simpliciorsquos proposal 10

24 Objective decoherence from the causal diamond 12

25 Global-local measure duality 17

26 Constructing a global multiverse from many causal diamond worlds 20

3 The many worlds of the census taker 24

31 Decoherence and recoherence 24

32 Failure to irreversibly decohere A limitation of finite systems 26

33 Sagredorsquos postulates 27

34 Irreversible decoherence and infinite repetition in the hat 33

35 Black hole complementarity and hat complementarity 36

36 The global multiverse in a hat 41

1 Introduction

According to an older view of quantum mechanics objective phenomena only occur

when an observation is made and as a result the wave function collapses A more

modern view called decoherence considers the effects of an inaccessible environment

that becomes entangled with the system of interest (including the observer) But at

what point precisely do the virtual realities described by a quantum mechanical wave

function turn into objective realities

This question is not about philosophy Without a precise form of decoherence

one cannot claim that anything really ldquohappenedrdquo including the specific outcomes of

experiments And without the ability to causally access an infinite number of pre-

cisely decohered outcomes one cannot reliably verify the probabilistic predictions of a

quantum-mechanical theory

ndash 1 ndash

The purpose of this paper is to argue that these questions may be resolved by

cosmology We will offer some principles that we believe are necessary for a consistent

interpretation of quantum mechanics and we will argue that eternal inflation is the

only cosmology which satisfies those principles There are two views of an eternally

inflating multiverse global (or parallel) vs local (or series) The parallel view is like

looking at a tree and seeing all its branches and twigs simultaneously The series view

is what is seen by an insect climbing from the base of the tree to a particular twig along

a specific route

In both the many-worlds interpretation of quantum mechanics and the multiverse

of eternal inflation the world is viewed as an unbounded collection of parallel universes

A view that has been expressed in the past by both of us is that there is no need to

add an additional layer of parallelism to the multiverse in order to interpret quantum

mechanics To put it succinctly the many-worlds and the multiverse are the same

thing [1]

Decoherence Decoherence1 explains why observers do not experience superpositions

of macroscopically distinct quantum states such as a superposition of an alive and a

dead cat The key insight is that macroscopic objects tend to quickly become entangled

with a large number of ldquoenvironmentalrdquo degrees of freedom E such as thermal photons

In practice these degrees of freedom cannot be monitored by the observer Whenever

a subsystem E is not monitored all expectation values behave as if the remaining

system is in a density matrix obtained by a partial trace over the Hilbert space of E

The density matrix will be diagonal in a preferred basis determined by the nature of

the interaction with the environment

As an example consider an isolated quantum system S with a two-dimensional

Hilbert space in the general state a|0〉S + b|1〉S Suppose a measurement takes place in

a small spacetime region which we may idealize as an event M By this we mean that

at M the system S interacts and becomes correlated with the pointer of an apparatus

A

(a|0〉S + b|1〉S)otimes |0〉A rarr a |0〉S otimes |0〉A + b |1〉S otimes |1〉A (11)

This process is unitary and is referred to as a pre-measurement

We assume that the apparatus is not a closed system (This is certainly the case

in practice for a macroscopic apparatus) Thus shortly afterwards (still at M in our

idealization) environmental degrees of freedom E scatter off of the apparatus and

become entangled with it By unitarity the system SAE as a whole remains in a pure

1For reviews see [2 3] For a pedagogical introduction see [4]

ndash 2 ndash

quantum state2

|ψ〉 = a |0〉S otimes |0〉A otimes |0〉E + b |1〉S otimes |1〉A otimes |1〉E (12)

We assume that the observer does not monitor the environment therefore he will

describe the state of SA by a density matrix obtained by a partial trace over the

Hilbert space factor representing the environment

ρSA = TrE|ψ〉〈ψ| (13)

This matrix is diagonal in the basis |0〉S otimes |0〉A |0〉S otimes |1〉A |1〉S otimes |0〉A |1〉S otimes |1〉A of

the Hilbert space of SA

ρSA = diag(|a|2 0 0 |b|2) (14)

This corresponds to a classical ensemble in which the pure state |0〉S otimes |0〉A has prob-

ability |a|2 and the state |1〉S otimes |1〉A has probability |b|2

Decoherence explains the ldquocollapse of the wave functionrdquo of the Copenhagen in-

terpretation as the non-unitary evolution from a pure to a mixed state resulting from

ignorance about an entangled subsystem E It also explains the very special quantum

states of macroscopic objects we experience as the elements of the basis in which the

density matrix ρSA is diagonal This preferred basis is picked out by the apparatus

configurations that scatter the environment into orthogonal states Because interac-

tions are usually local in space ρSA will be diagonal with respect to a basis consisting

of approximate position space eigenstates This explains why we perceive apparatus

states |0〉A (pointer up) or |1〉A (pointer down) but never the equally valid basis states

|plusmn〉A equiv 2minus12(|0〉Aplusmn|1〉A) which would correspond to superpositions of different pointer

positions

The entangled state obtained after premeasurement Eq (11) is a superposition

of two unentangled pure states or ldquobranchesrdquo In each branch the observer sees a

definite outcome |0〉 or |1〉 This in itself does not explain however why a definite

outcome is seen with respect to the basis |0〉 |1〉 rather than |+〉 |minus〉 Because

the decomposition of Eq (11) is not unique [5] the interaction with an inaccessible

environment and the resulting density matrix are essential to the selection of a preferred

basis of macroscopic states

Decoherence has two important limitations it is subjective and it is in principle

reversible This is a problem if we rely on decoherence for precise tests of quantum

2We could explicitly include an observer who becomes correlated to the apparatus through inter-

action with the environment resulting in an entangled pure state of the form a|0〉S otimes |0〉A otimes |0〉E otimes|0〉O + b|1〉S otimes |1〉A otimes |1〉E otimes |1〉O For notational simplicity we will subsume the observer into A

ndash 3 ndash

mechanical predictions We argue in Sec 2 that causal diamonds provide a natural

definition of environment in the multiverse leading to an observer-independent notion

of decoherent histories In Sec 3 we argue that these histories have precise irreversible

counterparts in the ldquohatrdquo-regions of the multiverse We now give a more detailed

overview of this paper

Outline In Sec 2 we address the first limitation of decoherence its subjectivity Be-

cause coherence is never lost in the full Hilbert space SAE the speed extent and

possible outcomes of decoherence depend on the definition of the environment E This

choice is made implicitly by an observer based on practical circumstances the envi-

ronment consists of degrees of freedom that have become entangled with the system

and apparatus but remain unobserved It is impractical for example to keep track of

every thermal photon emitted by a table of all of its interactions with light and air

particles and so on But if we did then we would find that the entire system SAE

behaves as a pure state |ψ〉 which may be a ldquocat staterdquo involving the superposition

of macroscopically different matter configurations Decoherence thus arises from the

description of the world by an observer who has access only to a subsystem To the

extent that the environment is defined by what a given observer cannot measure in

practice decoherence is subjective

The subjectivity of decoherence is not a problem as long as we are content to

explain our own experience ie that of an observer immersed in a much larger system

But the lack of any environment implies that decoherence cannot occur in a complete

unitary description of the whole universe It is possible that no such description exists

for our universe In Sec 21 we will argue however that causality places restrictions

on decoherence in much smaller regions in which the applicability of unitary quantum-

mechanical evolution seems beyond doubt

In Sec 22 we apply our analysis of decoherence and causality to eternal infla-

tion We will obtain a straightforward but perhaps surprising consequence in a global

description of an eternally inflating spacetime decoherence cannot occur so it is in-

consistent to imagine that pocket universes or vacuum bubbles nucleate at particular

locations and times In Sec 23 we discuss a number of attempts to rescue a unitary

global description and conclude that they do not succeed

In Sec 24 we review the ldquocausal diamondrdquo description of the multiverse The

causal diamond is the largest spacetime region that can be causally probed and it can

be thought of as the past light-cone from a point on the future conformal boundary

We argue that the causal diamond description leads to a natural observer-independent

choice of environment because its boundary is light-like it acts as a one-way membrane

and degrees of freedom that leave the diamond do not return except in very special

ndash 4 ndash

cases These degrees of freedom can be traced over leading to a branching tree of

causal diamond histories

Next we turn to the question of whether the global picture of the multiverse can

be recovered from the decoherent causal diamonds In Sec 25 we review a known

duality between the causal diamond and a particular foliation of the global geometry

known as light-cone time both give the same probabilities This duality took the

standard global picture as a starting point but in Sec 26 we reinterpret it as a way of

reconstructing the global viewpoint from the local one If the causal diamond histories

are the many-worlds this construction shows that the multiverse is the many-worlds

pieced together in a single geometry

In Sec 3 we turn to the second limitation associated with decoherence its re-

versibility Consider a causal diamond with finite maximal boundary area Amax En-

tropy bounds imply that such diamonds can be described by a Hilbert space with finite

dimension no greater than exp(Amax2) [67]3 This means that no observables in such

diamonds can be defined with infinite precision In Sec 31 and 32 we will discuss

another implication of this finiteness there is a tiny but nonzero probability that deco-

herence will be undone This means that the decoherent histories of causal diamonds

and the reconstruction of a global spacetime from such diamonds is not completely

exact

No matter how good an approximation is it is important to understand the precise

statement that it is an approximation to In Sec 33 we will develop two postulates

that should be satisfied by a fundamental quantum-mechanical theory if decoherence

is to be sharp and the associated probabilities operationally meaningful decoherence

must be irreversible and it must occur infinitely many times for a given experiment in

a single causally connected region

The string landscape contains supersymmetric vacua with exactly vanishing cosmo-

logical constant Causal diamonds which enter such vacua have infinite boundary area

at late times We argue in Sec 34 that in these ldquohatrdquo regions all our postulates can

be satisfied Exact observables can exist and decoherence by the mechanism of Sec 24

can be truly irreversible Moreover because the hat is a spatially open statistically

homogeneous universe anything that happens in the hat will happen infinitely many

times

In Sec 35 we review black hole complementarity and we conjecture an analogous

ldquohat complementarityrdquo for the multiverse It ensures that the approximate observables

and approximate decoherence of causal diamonds with finite area (Sec 24) have precise

counterparts in the hat In Sec 36 we propose a relation between the global multiverse

3This point has long been emphasized by Banks and Fischler [8ndash10]

ndash 5 ndash

M

E S+A

Figure 1 Decoherence and causality At the event M a macroscopic apparatus A becomes

correlated with a quantum system S Thereafter environmental degrees of freedom E interact

with the apparatus In practice an observer viewing the apparatus is ignorant of the exact

state of the environment and so must trace over this Hilbert space factor This results in a

mixed state which is diagonal in a particular ldquopointerrdquo basis picked out by the interaction

between E and A The state of the full system SAE however remains pure In particular

decoherence does not take place and no preferred bases arises in a complete description of

any region larger than the future lightcone of M

reconstruction of Sec 26 and the Census Taker cutoff [11] on the hat geometry

Two interesting papers have recently explored relations between the many-worlds

interpretation and the multiverse [12 13] The present work differs substantially in a

number of aspects Among them is the notion that causal diamonds provide a pre-

ferred environment for decoherence our view of the global multiverse as a patchwork

of decoherent causal diamonds our postulates requiring irreversible entanglement and

infinite repetition and the associated role we ascribe to hat regions of the multiverse

2 Building the multiverse from the many worlds of causal di-

amonds

21 Decoherence and causality

The decoherence mechanism reviewed above relies on ignoring the degrees of freedom

that a given observer fails to monitor which is fine if our goal is to explain the ex-

ndash 6 ndash

periences of that observer But this subjective viewpoint clashes with the impersonal

unitary description of large spacetime regionsmdashthe viewpoint usually adopted in cos-

mology We are free of course to pick any subsystem and trace over it But the

outcome will depend on this choice The usual choices implicitly involve locality but

not in a unique way

For example we might choose S to be an electron and E to be the inanimate

laboratory The systemrsquos wave function collapses when the electron becomes entangled

with some detector But we may also include in S everything out to the edge of the

solar system The environment is whatever is out beyond the orbit of Pluto In that

case the collapse of the system wavefunction cannot take place until a photon from the

detector has passed Plutorsquos orbit This would take about a five hours during which the

system wavefunction is coherent

In particular decoherence cannot occur in the complete quantum description of

any region larger than the future light-cone of the measurement event M (Fig 1) All

environmental degrees of freedom that could have become entangled with the apparatus

since the measurement took place must lie within this lightcone and hence are included

not traced over in a complete description of the state An example of such a region

is the whole universe ie any Cauchy surface to the future of M But at least at

sufficiently early times the future light-cone of M will be much smaller than the whole

universe Already on this scale the system SAE will be coherent

In our earlier example suppose that we measure the spin of an electron that is

initially prepared in a superposition of spin-up and spin-down a|0〉S + b|1〉S resulting

in the state |ψ〉 of Eq (12) A complete description of the solar system (defined as

the interior of a sphere the size of Plutorsquos orbit with a light-crossing time of about

10 hours) by a local quantum field theory contains every particle that could possibly

have interacted with the apparatus after the measurement for about 5 hours This

description would maintain the coherence of the macroscopic superpositions implicit in

the state |ψ〉 such as apparatus-up with apparatus-down until the first photons that

are entangled with the apparatus leave the solar system

Of course a detailed knowledge of the quantum state in such large regions is

unavailable to a realistic observer (Indeed if the region is larger than a cosmological

event horizon then its quantum state is cannot be probed at all without violating

causality) Yet our theoretical description of matter fields in spacetime retains in

principle all degrees of freedom and full coherence of the quantum state In theoretical

cosmology this can lead to inconsistencies if we describe regions that are larger than the

future light-cones of events that we nevertheless treat as decohered We now consider

an important example

ndash 7 ndash

22 Failure to decohere A problem with the global multiverse

The above analysis undermines what we will call the ldquostandard global picturerdquo of an

eternally inflating spacetime Consider an effective potential containing at least one

inflating false vacuum ie a metastable de Sitter vacuum with decay rate much less

than one decay per Hubble volume and Hubble time We will also assume that there

is at least one terminal vacuum with nonpositive cosmological constant (The string

theory landscape is believed to have at least 10100primes of vacua of both types [14ndash17])

According to the standard description of eternal inflation an inflating vacuum nu-

cleates bubble-universes in a statistical manner similar to the way superheated water

nucleates bubbles of steam That process is described by classical stochastic production

of bubbles which occurs randomly but the randomness is classical The bubbles nucle-

ate at definite locations and coherent quantum mechanical interference plays no role

The conventional description of eternal inflation similarly based on classical stochastic

processes However this picture is not consistent with a complete quantum-mechanical

description of a global region of the multiverse

To explain why this is so consider the future domain of dependence D(Σ0) of a

sufficiently large hypersurface Σ0 which need not be a Cauchy surface D(Σ0) consists

of all events that can be predicted from data on Σ0 see Fig 2 If Σ0 contains suffi-

ciently large and long-lived metastable de Sitter regions then bubbles of vacua of lower

energy do not consume the parent de Sitter vacua in which they nucleate [18] Hence

the de Sitter vacua are said to inflate eternally producing an unbounded number of

bubble universes The resulting spacetime is said to have the structure shown in the

conformal diagram in Fig 2 with bubbles nucleating at definite spacetime events The

future conformal boundary is spacelike in regions with negative cosmological constant

corresponding to a local big crunch The boundary contains null ldquohatsrdquo in regions

occupied by vacua with Λ = 0

But this picture does not arise in a complete quantum description of D(Σ0) The

future light-cones of events at late times are much smaller than D(Σ0) In any state

that describes the entire spacetime region D(Σ0) decoherence can only take place at

the margin of D(Σ0) (shown light shaded in Fig 2) in the region from which particles

can escape into the complement of D(Σ0) in the full spacetime No decoherence can

take place in the infinite spacetime region defined by the past domain of dependence

of the future boundary of D(Σ0) In this region quantum evolution remains coherent

even if it results in the superposition of macroscopically distinct matter or spacetime

configurations

An important example is the superposition of vacuum decays taking place at dif-

ferent places Without decoherence it makes no sense to say that bubbles nucleate at

ndash 8 ndash

future boundary

Σ0

Figure 2 The future domain of dependence D(Σ0) (light or dark shaded) is the spacetime

region that can be predicted from data on the timeslice Σ0 If the future conformal boundary

contains spacelike portions as in eternal inflation or inside a black hole then the future

light-cones of events in the dark shaded region remain entirely within D(Σ0) Pure quantum

states do not decohere in this region in a complete description of D(Σ0) This is true even for

states that involve macroscopic superpositions such as the locations of pocket universes in

eternal inflation (dashed lines) calling into question the self-consistency of the global picture

of eternal inflation

particular times and locations rather a wavefunction with initial support only in the

parent vacuum develops into a superposition of parent and daughter vacua Bubbles

nucleating at all places at times are ldquoquantum superimposedrdquo With the gravitational

backreaction included the metric too would remain in a quantum-mechanical super-

position This contradicts the standard global picture of eternal inflation in which

domain walls vacua and the spacetime metric take on definite values as if drawn from

a density matrix obtained by tracing over some degrees of freedom and as if the inter-

action with these degrees of freedom had picked out a preferred basis that eliminates

the quantum superposition of bubbles and vacua

Let us quickly get rid of one red herring Can the standard geometry of eternal

inflation be recovered by using so-called semi-classical gravity in which the metric is

sourced by the expectation value of the energy-momentum tensor

Gmicroν = 8π〈Tmicroν〉 (21)

This does not work because the matter quantum fields would still remain coherent At

the level of the quantum fields the wavefunction initially has support only in the false

vacuum Over time it evolves to a superposition of the false vacuum (with decreasing

amplitude) with the true vacuum (with increasing amplitude) plus a superposition

of expanding and colliding domain walls This state is quite complicated but the

expectation value of its stress tensor should remain spatially homogeneous if it was so

initially The net effect over time would be a continuous conversion of vacuum energy

into ordinary matter or radiation (from the collision of bubbles and motion of the scalar

field) By Eq (21) the geometry spacetime would respond to the homogeneous glide

ndash 9 ndash

of the vacuum energy to negative values This would result in a global crunch after

finite time in stark contrast to the standard picture of global eternal inflation In

any case it seems implausible that semi-classical gravity should apply in a situation in

which coherent branches of the wavefunction have radically different gravitational back-

reaction The AdSCFT correspondence provides an explicit counterexample since the

superposition of two CFT states that correspond to different classical geometries must

correspond to a quantum superposition of the two metrics

The conclusion that we come to from these considerations is not that the global

multiverse is meaningless but that the parallel view should not be implemented by

unitary quantum mechanics But is there an alternative Can the standard global

picture be recovered by considering an observer who has access only to some of the

degrees of freedom of the multiverse and appealing to decoherence We debate this

question in the following section

23 Simpliciorsquos proposal

Simplicio and Sagredo have studied Sections 21 and 22 supplied to them by Salviati

They meet at Sagredorsquos house for a discussion

Simplicio You have convinced me that a complete description of eternal inflation

by unitary quantum evolution on global slices will not lead to a picture in which bubbles

form at definite places and times But all I need is an observer somewhere Then I

can take this observerrsquos point of view and trace over the degrees of freedom that are

inaccessible to him This decoheres events such as bubble nucleations in the entire

global multiverse It actually helps that some regions are causally disconnected from the

observer this makes his environmentmdashthe degrees of freedom he fails to accessmdashreally

huge

Sagredo An interesting idea But you seem to include everything outside the

observerrsquos horizon region in what you call the environment Once you trace over it it

is gone from your description and you could not possibly recover a global spacetime

Simplicio Your objection is valid but it also shows me how to fix my proposal

The observer should only trace over environmental degrees in his own horizon Deco-

herence is very efficient so this should suffice

Sagredo I wonder what would happen if there were two observers in widely

separated regions If one observerrsquos environment is enough to decohere the whole

universe which one should we pick

Simplicio I have not done a calculation but it seems to me that it shouldnrsquot

matter The outcome of an experiment by one of the observers should be the same no

matter which observerrsquos environment I trace over That is certainly how it works when

you and I both stare at the same apparatus

ndash 10 ndash

future boundary

Σ0

P O

Figure 3 Environmental degrees of freedom entangled with an observer at O remain within

the causal future of the causal past of O J+[Jminus(O)] (cyanshaded) They are not entangled

with distant regions of the multiverse Tracing over them will not lead to decoherence of

a bubble nucleated at P for example and hence will fail to reproduce the standard global

picture of eternal inflation

Sagredo Something is different about the multiverse When you and I both

observe Salviati we all become correlated by interactions with a common environment

But how does an observer in one horizon volume become correlated with an object in

another horizon volume far away

Salviati Sagredo you hit the nail on the head Decoherence requires the inter-

action of environmental degrees of freedom with the apparatus and the observer This

entangles them and it leads to a density matrix once the environment is ignored by the

observer But an observer cannot have interacted with degrees of freedom that were

never present in his past light-cone

Sagredo Thank you for articulating so clearly what to me was only a vague

concern Simplicio you look puzzled so let me summarize our objection in my own

words You proposed a method for obtaining the standard global picture of eternal

inflation you claim that we need only identify an arbitrary observer in the multiverse

and trace over his environment If we defined the environment as all degrees of freedom

the observer fails to monitor then it would include the causally disconnected regions

outside his horizon With this definition these regions will disappear entirely from your

description in conflict with the global picture So we agreed to define the environment

as the degrees of freedom that have interacted with the observer and which he cannot

access in practice But in this case the environment includes no degrees of freedom

outside the causal future of the observerrsquos causal past I have drawn this region in

Fig 3 But tracing over an environment can only decohere degrees of freedom that it

is entangled with In this case it can decohere some events that lie in the observerrsquos

past light-cone But it cannot affect quantum coherence in far-away horizon regions

because the environment you have picked is not entangled with these regions In those

ndash 11 ndash

regions bubble walls and vacua will remain in superposition which again conflicts with

the standard global picture of eternal inflation

Simplicio I see that my idea still has some problems I will need to identify more

than one observer-environment pair In fact if I wish to preserve the global picture

of the multiverse I will have to assume that an observer is present in every horizon

volume at all times Otherwise there will be horizon regions where no one is around

to decide which degrees of freedom are hard to keep track of so there is no way to

identify and trace over an environment In such regions bubbles would not form at

particular places and times in conflict with the standard global picture

Sagredo But this assumption is clearly violated in many landscape models Most

de Sitter vacua have large cosmological constant so that a single horizon volume is too

small to contain the large number of degrees of freedom required for an observer And

regions with small vacuum energy may be very long lived so the corresponding bubbles

contain many horizon volumes that are completely empty Irsquom afraid Simplicio that

your efforts to rescue the global multiverse are destined to fail

Salviati Why donrsquot we back up a little and return to Simpliciorsquos initial sug-

gestion Sagredo you objected that everything outside an observerrsquos horizon would

naturally be part of his environment and would be gone from our description if we

trace over it

Sagredo which means that the whole global description would be gone

Salviati but why is that a problem No observer inside the universe can ever

see more than what is in their past light-cone at late times or more precisely in their

causal diamond We may not be able to recover the global picture by tracing over

the region behind an observerrsquos horizon but the same procedure might well achieve

decoherence in the region the observer can actually access In fact we donrsquot even

need an actual observer we can get decoherence by tracing over degrees of freedom

that leave the causal horizon of any worldline This will allow us to say that a bubble

formed in one place and not another So why donrsquot we give up on the global description

for a moment Later on we can check whether a global picture can be recovered in

some way from the decoherent causal diamonds

Salviati hands out Sections 24ndash26

24 Objective decoherence from the causal diamond

If Hawking radiation contains the full information about the quantum state of a star

that collapsed to form a black hole then there is an apparent paradox The star is

located inside the black hole at spacelike separation from the Hawking cloud hence two

copies of the original quantum information are present simultaneously The xeroxing of

quantum information however conflicts with the linearity of quantum mechanics [19]

ndash 12 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

The purpose of this paper is to argue that these questions may be resolved by

cosmology We will offer some principles that we believe are necessary for a consistent

interpretation of quantum mechanics and we will argue that eternal inflation is the

only cosmology which satisfies those principles There are two views of an eternally

inflating multiverse global (or parallel) vs local (or series) The parallel view is like

looking at a tree and seeing all its branches and twigs simultaneously The series view

is what is seen by an insect climbing from the base of the tree to a particular twig along

a specific route

In both the many-worlds interpretation of quantum mechanics and the multiverse

of eternal inflation the world is viewed as an unbounded collection of parallel universes

A view that has been expressed in the past by both of us is that there is no need to

add an additional layer of parallelism to the multiverse in order to interpret quantum

mechanics To put it succinctly the many-worlds and the multiverse are the same

thing [1]

Decoherence Decoherence1 explains why observers do not experience superpositions

of macroscopically distinct quantum states such as a superposition of an alive and a

dead cat The key insight is that macroscopic objects tend to quickly become entangled

with a large number of ldquoenvironmentalrdquo degrees of freedom E such as thermal photons

In practice these degrees of freedom cannot be monitored by the observer Whenever

a subsystem E is not monitored all expectation values behave as if the remaining

system is in a density matrix obtained by a partial trace over the Hilbert space of E

The density matrix will be diagonal in a preferred basis determined by the nature of

the interaction with the environment

As an example consider an isolated quantum system S with a two-dimensional

Hilbert space in the general state a|0〉S + b|1〉S Suppose a measurement takes place in

a small spacetime region which we may idealize as an event M By this we mean that

at M the system S interacts and becomes correlated with the pointer of an apparatus

A

(a|0〉S + b|1〉S)otimes |0〉A rarr a |0〉S otimes |0〉A + b |1〉S otimes |1〉A (11)

This process is unitary and is referred to as a pre-measurement

We assume that the apparatus is not a closed system (This is certainly the case

in practice for a macroscopic apparatus) Thus shortly afterwards (still at M in our

idealization) environmental degrees of freedom E scatter off of the apparatus and

become entangled with it By unitarity the system SAE as a whole remains in a pure

1For reviews see [2 3] For a pedagogical introduction see [4]

ndash 2 ndash

quantum state2

|ψ〉 = a |0〉S otimes |0〉A otimes |0〉E + b |1〉S otimes |1〉A otimes |1〉E (12)

We assume that the observer does not monitor the environment therefore he will

describe the state of SA by a density matrix obtained by a partial trace over the

Hilbert space factor representing the environment

ρSA = TrE|ψ〉〈ψ| (13)

This matrix is diagonal in the basis |0〉S otimes |0〉A |0〉S otimes |1〉A |1〉S otimes |0〉A |1〉S otimes |1〉A of

the Hilbert space of SA

ρSA = diag(|a|2 0 0 |b|2) (14)

This corresponds to a classical ensemble in which the pure state |0〉S otimes |0〉A has prob-

ability |a|2 and the state |1〉S otimes |1〉A has probability |b|2

Decoherence explains the ldquocollapse of the wave functionrdquo of the Copenhagen in-

terpretation as the non-unitary evolution from a pure to a mixed state resulting from

ignorance about an entangled subsystem E It also explains the very special quantum

states of macroscopic objects we experience as the elements of the basis in which the

density matrix ρSA is diagonal This preferred basis is picked out by the apparatus

configurations that scatter the environment into orthogonal states Because interac-

tions are usually local in space ρSA will be diagonal with respect to a basis consisting

of approximate position space eigenstates This explains why we perceive apparatus

states |0〉A (pointer up) or |1〉A (pointer down) but never the equally valid basis states

|plusmn〉A equiv 2minus12(|0〉Aplusmn|1〉A) which would correspond to superpositions of different pointer

positions

The entangled state obtained after premeasurement Eq (11) is a superposition

of two unentangled pure states or ldquobranchesrdquo In each branch the observer sees a

definite outcome |0〉 or |1〉 This in itself does not explain however why a definite

outcome is seen with respect to the basis |0〉 |1〉 rather than |+〉 |minus〉 Because

the decomposition of Eq (11) is not unique [5] the interaction with an inaccessible

environment and the resulting density matrix are essential to the selection of a preferred

basis of macroscopic states

Decoherence has two important limitations it is subjective and it is in principle

reversible This is a problem if we rely on decoherence for precise tests of quantum

2We could explicitly include an observer who becomes correlated to the apparatus through inter-

action with the environment resulting in an entangled pure state of the form a|0〉S otimes |0〉A otimes |0〉E otimes|0〉O + b|1〉S otimes |1〉A otimes |1〉E otimes |1〉O For notational simplicity we will subsume the observer into A

ndash 3 ndash

mechanical predictions We argue in Sec 2 that causal diamonds provide a natural

definition of environment in the multiverse leading to an observer-independent notion

of decoherent histories In Sec 3 we argue that these histories have precise irreversible

counterparts in the ldquohatrdquo-regions of the multiverse We now give a more detailed

overview of this paper

Outline In Sec 2 we address the first limitation of decoherence its subjectivity Be-

cause coherence is never lost in the full Hilbert space SAE the speed extent and

possible outcomes of decoherence depend on the definition of the environment E This

choice is made implicitly by an observer based on practical circumstances the envi-

ronment consists of degrees of freedom that have become entangled with the system

and apparatus but remain unobserved It is impractical for example to keep track of

every thermal photon emitted by a table of all of its interactions with light and air

particles and so on But if we did then we would find that the entire system SAE

behaves as a pure state |ψ〉 which may be a ldquocat staterdquo involving the superposition

of macroscopically different matter configurations Decoherence thus arises from the

description of the world by an observer who has access only to a subsystem To the

extent that the environment is defined by what a given observer cannot measure in

practice decoherence is subjective

The subjectivity of decoherence is not a problem as long as we are content to

explain our own experience ie that of an observer immersed in a much larger system

But the lack of any environment implies that decoherence cannot occur in a complete

unitary description of the whole universe It is possible that no such description exists

for our universe In Sec 21 we will argue however that causality places restrictions

on decoherence in much smaller regions in which the applicability of unitary quantum-

mechanical evolution seems beyond doubt

In Sec 22 we apply our analysis of decoherence and causality to eternal infla-

tion We will obtain a straightforward but perhaps surprising consequence in a global

description of an eternally inflating spacetime decoherence cannot occur so it is in-

consistent to imagine that pocket universes or vacuum bubbles nucleate at particular

locations and times In Sec 23 we discuss a number of attempts to rescue a unitary

global description and conclude that they do not succeed

In Sec 24 we review the ldquocausal diamondrdquo description of the multiverse The

causal diamond is the largest spacetime region that can be causally probed and it can

be thought of as the past light-cone from a point on the future conformal boundary

We argue that the causal diamond description leads to a natural observer-independent

choice of environment because its boundary is light-like it acts as a one-way membrane

and degrees of freedom that leave the diamond do not return except in very special

ndash 4 ndash

cases These degrees of freedom can be traced over leading to a branching tree of

causal diamond histories

Next we turn to the question of whether the global picture of the multiverse can

be recovered from the decoherent causal diamonds In Sec 25 we review a known

duality between the causal diamond and a particular foliation of the global geometry

known as light-cone time both give the same probabilities This duality took the

standard global picture as a starting point but in Sec 26 we reinterpret it as a way of

reconstructing the global viewpoint from the local one If the causal diamond histories

are the many-worlds this construction shows that the multiverse is the many-worlds

pieced together in a single geometry

In Sec 3 we turn to the second limitation associated with decoherence its re-

versibility Consider a causal diamond with finite maximal boundary area Amax En-

tropy bounds imply that such diamonds can be described by a Hilbert space with finite

dimension no greater than exp(Amax2) [67]3 This means that no observables in such

diamonds can be defined with infinite precision In Sec 31 and 32 we will discuss

another implication of this finiteness there is a tiny but nonzero probability that deco-

herence will be undone This means that the decoherent histories of causal diamonds

and the reconstruction of a global spacetime from such diamonds is not completely

exact

No matter how good an approximation is it is important to understand the precise

statement that it is an approximation to In Sec 33 we will develop two postulates

that should be satisfied by a fundamental quantum-mechanical theory if decoherence

is to be sharp and the associated probabilities operationally meaningful decoherence

must be irreversible and it must occur infinitely many times for a given experiment in

a single causally connected region

The string landscape contains supersymmetric vacua with exactly vanishing cosmo-

logical constant Causal diamonds which enter such vacua have infinite boundary area

at late times We argue in Sec 34 that in these ldquohatrdquo regions all our postulates can

be satisfied Exact observables can exist and decoherence by the mechanism of Sec 24

can be truly irreversible Moreover because the hat is a spatially open statistically

homogeneous universe anything that happens in the hat will happen infinitely many

times

In Sec 35 we review black hole complementarity and we conjecture an analogous

ldquohat complementarityrdquo for the multiverse It ensures that the approximate observables

and approximate decoherence of causal diamonds with finite area (Sec 24) have precise

counterparts in the hat In Sec 36 we propose a relation between the global multiverse

3This point has long been emphasized by Banks and Fischler [8ndash10]

ndash 5 ndash

M

E S+A

Figure 1 Decoherence and causality At the event M a macroscopic apparatus A becomes

correlated with a quantum system S Thereafter environmental degrees of freedom E interact

with the apparatus In practice an observer viewing the apparatus is ignorant of the exact

state of the environment and so must trace over this Hilbert space factor This results in a

mixed state which is diagonal in a particular ldquopointerrdquo basis picked out by the interaction

between E and A The state of the full system SAE however remains pure In particular

decoherence does not take place and no preferred bases arises in a complete description of

any region larger than the future lightcone of M

reconstruction of Sec 26 and the Census Taker cutoff [11] on the hat geometry

Two interesting papers have recently explored relations between the many-worlds

interpretation and the multiverse [12 13] The present work differs substantially in a

number of aspects Among them is the notion that causal diamonds provide a pre-

ferred environment for decoherence our view of the global multiverse as a patchwork

of decoherent causal diamonds our postulates requiring irreversible entanglement and

infinite repetition and the associated role we ascribe to hat regions of the multiverse

2 Building the multiverse from the many worlds of causal di-

amonds

21 Decoherence and causality

The decoherence mechanism reviewed above relies on ignoring the degrees of freedom

that a given observer fails to monitor which is fine if our goal is to explain the ex-

ndash 6 ndash

periences of that observer But this subjective viewpoint clashes with the impersonal

unitary description of large spacetime regionsmdashthe viewpoint usually adopted in cos-

mology We are free of course to pick any subsystem and trace over it But the

outcome will depend on this choice The usual choices implicitly involve locality but

not in a unique way

For example we might choose S to be an electron and E to be the inanimate

laboratory The systemrsquos wave function collapses when the electron becomes entangled

with some detector But we may also include in S everything out to the edge of the

solar system The environment is whatever is out beyond the orbit of Pluto In that

case the collapse of the system wavefunction cannot take place until a photon from the

detector has passed Plutorsquos orbit This would take about a five hours during which the

system wavefunction is coherent

In particular decoherence cannot occur in the complete quantum description of

any region larger than the future light-cone of the measurement event M (Fig 1) All

environmental degrees of freedom that could have become entangled with the apparatus

since the measurement took place must lie within this lightcone and hence are included

not traced over in a complete description of the state An example of such a region

is the whole universe ie any Cauchy surface to the future of M But at least at

sufficiently early times the future light-cone of M will be much smaller than the whole

universe Already on this scale the system SAE will be coherent

In our earlier example suppose that we measure the spin of an electron that is

initially prepared in a superposition of spin-up and spin-down a|0〉S + b|1〉S resulting

in the state |ψ〉 of Eq (12) A complete description of the solar system (defined as

the interior of a sphere the size of Plutorsquos orbit with a light-crossing time of about

10 hours) by a local quantum field theory contains every particle that could possibly

have interacted with the apparatus after the measurement for about 5 hours This

description would maintain the coherence of the macroscopic superpositions implicit in

the state |ψ〉 such as apparatus-up with apparatus-down until the first photons that

are entangled with the apparatus leave the solar system

Of course a detailed knowledge of the quantum state in such large regions is

unavailable to a realistic observer (Indeed if the region is larger than a cosmological

event horizon then its quantum state is cannot be probed at all without violating

causality) Yet our theoretical description of matter fields in spacetime retains in

principle all degrees of freedom and full coherence of the quantum state In theoretical

cosmology this can lead to inconsistencies if we describe regions that are larger than the

future light-cones of events that we nevertheless treat as decohered We now consider

an important example

ndash 7 ndash

22 Failure to decohere A problem with the global multiverse

The above analysis undermines what we will call the ldquostandard global picturerdquo of an

eternally inflating spacetime Consider an effective potential containing at least one

inflating false vacuum ie a metastable de Sitter vacuum with decay rate much less

than one decay per Hubble volume and Hubble time We will also assume that there

is at least one terminal vacuum with nonpositive cosmological constant (The string

theory landscape is believed to have at least 10100primes of vacua of both types [14ndash17])

According to the standard description of eternal inflation an inflating vacuum nu-

cleates bubble-universes in a statistical manner similar to the way superheated water

nucleates bubbles of steam That process is described by classical stochastic production

of bubbles which occurs randomly but the randomness is classical The bubbles nucle-

ate at definite locations and coherent quantum mechanical interference plays no role

The conventional description of eternal inflation similarly based on classical stochastic

processes However this picture is not consistent with a complete quantum-mechanical

description of a global region of the multiverse

To explain why this is so consider the future domain of dependence D(Σ0) of a

sufficiently large hypersurface Σ0 which need not be a Cauchy surface D(Σ0) consists

of all events that can be predicted from data on Σ0 see Fig 2 If Σ0 contains suffi-

ciently large and long-lived metastable de Sitter regions then bubbles of vacua of lower

energy do not consume the parent de Sitter vacua in which they nucleate [18] Hence

the de Sitter vacua are said to inflate eternally producing an unbounded number of

bubble universes The resulting spacetime is said to have the structure shown in the

conformal diagram in Fig 2 with bubbles nucleating at definite spacetime events The

future conformal boundary is spacelike in regions with negative cosmological constant

corresponding to a local big crunch The boundary contains null ldquohatsrdquo in regions

occupied by vacua with Λ = 0

But this picture does not arise in a complete quantum description of D(Σ0) The

future light-cones of events at late times are much smaller than D(Σ0) In any state

that describes the entire spacetime region D(Σ0) decoherence can only take place at

the margin of D(Σ0) (shown light shaded in Fig 2) in the region from which particles

can escape into the complement of D(Σ0) in the full spacetime No decoherence can

take place in the infinite spacetime region defined by the past domain of dependence

of the future boundary of D(Σ0) In this region quantum evolution remains coherent

even if it results in the superposition of macroscopically distinct matter or spacetime

configurations

An important example is the superposition of vacuum decays taking place at dif-

ferent places Without decoherence it makes no sense to say that bubbles nucleate at

ndash 8 ndash

future boundary

Σ0

Figure 2 The future domain of dependence D(Σ0) (light or dark shaded) is the spacetime

region that can be predicted from data on the timeslice Σ0 If the future conformal boundary

contains spacelike portions as in eternal inflation or inside a black hole then the future

light-cones of events in the dark shaded region remain entirely within D(Σ0) Pure quantum

states do not decohere in this region in a complete description of D(Σ0) This is true even for

states that involve macroscopic superpositions such as the locations of pocket universes in

eternal inflation (dashed lines) calling into question the self-consistency of the global picture

of eternal inflation

particular times and locations rather a wavefunction with initial support only in the

parent vacuum develops into a superposition of parent and daughter vacua Bubbles

nucleating at all places at times are ldquoquantum superimposedrdquo With the gravitational

backreaction included the metric too would remain in a quantum-mechanical super-

position This contradicts the standard global picture of eternal inflation in which

domain walls vacua and the spacetime metric take on definite values as if drawn from

a density matrix obtained by tracing over some degrees of freedom and as if the inter-

action with these degrees of freedom had picked out a preferred basis that eliminates

the quantum superposition of bubbles and vacua

Let us quickly get rid of one red herring Can the standard geometry of eternal

inflation be recovered by using so-called semi-classical gravity in which the metric is

sourced by the expectation value of the energy-momentum tensor

Gmicroν = 8π〈Tmicroν〉 (21)

This does not work because the matter quantum fields would still remain coherent At

the level of the quantum fields the wavefunction initially has support only in the false

vacuum Over time it evolves to a superposition of the false vacuum (with decreasing

amplitude) with the true vacuum (with increasing amplitude) plus a superposition

of expanding and colliding domain walls This state is quite complicated but the

expectation value of its stress tensor should remain spatially homogeneous if it was so

initially The net effect over time would be a continuous conversion of vacuum energy

into ordinary matter or radiation (from the collision of bubbles and motion of the scalar

field) By Eq (21) the geometry spacetime would respond to the homogeneous glide

ndash 9 ndash

of the vacuum energy to negative values This would result in a global crunch after

finite time in stark contrast to the standard picture of global eternal inflation In

any case it seems implausible that semi-classical gravity should apply in a situation in

which coherent branches of the wavefunction have radically different gravitational back-

reaction The AdSCFT correspondence provides an explicit counterexample since the

superposition of two CFT states that correspond to different classical geometries must

correspond to a quantum superposition of the two metrics

The conclusion that we come to from these considerations is not that the global

multiverse is meaningless but that the parallel view should not be implemented by

unitary quantum mechanics But is there an alternative Can the standard global

picture be recovered by considering an observer who has access only to some of the

degrees of freedom of the multiverse and appealing to decoherence We debate this

question in the following section

23 Simpliciorsquos proposal

Simplicio and Sagredo have studied Sections 21 and 22 supplied to them by Salviati

They meet at Sagredorsquos house for a discussion

Simplicio You have convinced me that a complete description of eternal inflation

by unitary quantum evolution on global slices will not lead to a picture in which bubbles

form at definite places and times But all I need is an observer somewhere Then I

can take this observerrsquos point of view and trace over the degrees of freedom that are

inaccessible to him This decoheres events such as bubble nucleations in the entire

global multiverse It actually helps that some regions are causally disconnected from the

observer this makes his environmentmdashthe degrees of freedom he fails to accessmdashreally

huge

Sagredo An interesting idea But you seem to include everything outside the

observerrsquos horizon region in what you call the environment Once you trace over it it

is gone from your description and you could not possibly recover a global spacetime

Simplicio Your objection is valid but it also shows me how to fix my proposal

The observer should only trace over environmental degrees in his own horizon Deco-

herence is very efficient so this should suffice

Sagredo I wonder what would happen if there were two observers in widely

separated regions If one observerrsquos environment is enough to decohere the whole

universe which one should we pick

Simplicio I have not done a calculation but it seems to me that it shouldnrsquot

matter The outcome of an experiment by one of the observers should be the same no

matter which observerrsquos environment I trace over That is certainly how it works when

you and I both stare at the same apparatus

ndash 10 ndash

future boundary

Σ0

P O

Figure 3 Environmental degrees of freedom entangled with an observer at O remain within

the causal future of the causal past of O J+[Jminus(O)] (cyanshaded) They are not entangled

with distant regions of the multiverse Tracing over them will not lead to decoherence of

a bubble nucleated at P for example and hence will fail to reproduce the standard global

picture of eternal inflation

Sagredo Something is different about the multiverse When you and I both

observe Salviati we all become correlated by interactions with a common environment

But how does an observer in one horizon volume become correlated with an object in

another horizon volume far away

Salviati Sagredo you hit the nail on the head Decoherence requires the inter-

action of environmental degrees of freedom with the apparatus and the observer This

entangles them and it leads to a density matrix once the environment is ignored by the

observer But an observer cannot have interacted with degrees of freedom that were

never present in his past light-cone

Sagredo Thank you for articulating so clearly what to me was only a vague

concern Simplicio you look puzzled so let me summarize our objection in my own

words You proposed a method for obtaining the standard global picture of eternal

inflation you claim that we need only identify an arbitrary observer in the multiverse

and trace over his environment If we defined the environment as all degrees of freedom

the observer fails to monitor then it would include the causally disconnected regions

outside his horizon With this definition these regions will disappear entirely from your

description in conflict with the global picture So we agreed to define the environment

as the degrees of freedom that have interacted with the observer and which he cannot

access in practice But in this case the environment includes no degrees of freedom

outside the causal future of the observerrsquos causal past I have drawn this region in

Fig 3 But tracing over an environment can only decohere degrees of freedom that it

is entangled with In this case it can decohere some events that lie in the observerrsquos

past light-cone But it cannot affect quantum coherence in far-away horizon regions

because the environment you have picked is not entangled with these regions In those

ndash 11 ndash

regions bubble walls and vacua will remain in superposition which again conflicts with

the standard global picture of eternal inflation

Simplicio I see that my idea still has some problems I will need to identify more

than one observer-environment pair In fact if I wish to preserve the global picture

of the multiverse I will have to assume that an observer is present in every horizon

volume at all times Otherwise there will be horizon regions where no one is around

to decide which degrees of freedom are hard to keep track of so there is no way to

identify and trace over an environment In such regions bubbles would not form at

particular places and times in conflict with the standard global picture

Sagredo But this assumption is clearly violated in many landscape models Most

de Sitter vacua have large cosmological constant so that a single horizon volume is too

small to contain the large number of degrees of freedom required for an observer And

regions with small vacuum energy may be very long lived so the corresponding bubbles

contain many horizon volumes that are completely empty Irsquom afraid Simplicio that

your efforts to rescue the global multiverse are destined to fail

Salviati Why donrsquot we back up a little and return to Simpliciorsquos initial sug-

gestion Sagredo you objected that everything outside an observerrsquos horizon would

naturally be part of his environment and would be gone from our description if we

trace over it

Sagredo which means that the whole global description would be gone

Salviati but why is that a problem No observer inside the universe can ever

see more than what is in their past light-cone at late times or more precisely in their

causal diamond We may not be able to recover the global picture by tracing over

the region behind an observerrsquos horizon but the same procedure might well achieve

decoherence in the region the observer can actually access In fact we donrsquot even

need an actual observer we can get decoherence by tracing over degrees of freedom

that leave the causal horizon of any worldline This will allow us to say that a bubble

formed in one place and not another So why donrsquot we give up on the global description

for a moment Later on we can check whether a global picture can be recovered in

some way from the decoherent causal diamonds

Salviati hands out Sections 24ndash26

24 Objective decoherence from the causal diamond

If Hawking radiation contains the full information about the quantum state of a star

that collapsed to form a black hole then there is an apparent paradox The star is

located inside the black hole at spacelike separation from the Hawking cloud hence two

copies of the original quantum information are present simultaneously The xeroxing of

quantum information however conflicts with the linearity of quantum mechanics [19]

ndash 12 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

quantum state2

|ψ〉 = a |0〉S otimes |0〉A otimes |0〉E + b |1〉S otimes |1〉A otimes |1〉E (12)

We assume that the observer does not monitor the environment therefore he will

describe the state of SA by a density matrix obtained by a partial trace over the

Hilbert space factor representing the environment

ρSA = TrE|ψ〉〈ψ| (13)

This matrix is diagonal in the basis |0〉S otimes |0〉A |0〉S otimes |1〉A |1〉S otimes |0〉A |1〉S otimes |1〉A of

the Hilbert space of SA

ρSA = diag(|a|2 0 0 |b|2) (14)

This corresponds to a classical ensemble in which the pure state |0〉S otimes |0〉A has prob-

ability |a|2 and the state |1〉S otimes |1〉A has probability |b|2

Decoherence explains the ldquocollapse of the wave functionrdquo of the Copenhagen in-

terpretation as the non-unitary evolution from a pure to a mixed state resulting from

ignorance about an entangled subsystem E It also explains the very special quantum

states of macroscopic objects we experience as the elements of the basis in which the

density matrix ρSA is diagonal This preferred basis is picked out by the apparatus

configurations that scatter the environment into orthogonal states Because interac-

tions are usually local in space ρSA will be diagonal with respect to a basis consisting

of approximate position space eigenstates This explains why we perceive apparatus

states |0〉A (pointer up) or |1〉A (pointer down) but never the equally valid basis states

|plusmn〉A equiv 2minus12(|0〉Aplusmn|1〉A) which would correspond to superpositions of different pointer

positions

The entangled state obtained after premeasurement Eq (11) is a superposition

of two unentangled pure states or ldquobranchesrdquo In each branch the observer sees a

definite outcome |0〉 or |1〉 This in itself does not explain however why a definite

outcome is seen with respect to the basis |0〉 |1〉 rather than |+〉 |minus〉 Because

the decomposition of Eq (11) is not unique [5] the interaction with an inaccessible

environment and the resulting density matrix are essential to the selection of a preferred

basis of macroscopic states

Decoherence has two important limitations it is subjective and it is in principle

reversible This is a problem if we rely on decoherence for precise tests of quantum

2We could explicitly include an observer who becomes correlated to the apparatus through inter-

action with the environment resulting in an entangled pure state of the form a|0〉S otimes |0〉A otimes |0〉E otimes|0〉O + b|1〉S otimes |1〉A otimes |1〉E otimes |1〉O For notational simplicity we will subsume the observer into A

ndash 3 ndash

mechanical predictions We argue in Sec 2 that causal diamonds provide a natural

definition of environment in the multiverse leading to an observer-independent notion

of decoherent histories In Sec 3 we argue that these histories have precise irreversible

counterparts in the ldquohatrdquo-regions of the multiverse We now give a more detailed

overview of this paper

Outline In Sec 2 we address the first limitation of decoherence its subjectivity Be-

cause coherence is never lost in the full Hilbert space SAE the speed extent and

possible outcomes of decoherence depend on the definition of the environment E This

choice is made implicitly by an observer based on practical circumstances the envi-

ronment consists of degrees of freedom that have become entangled with the system

and apparatus but remain unobserved It is impractical for example to keep track of

every thermal photon emitted by a table of all of its interactions with light and air

particles and so on But if we did then we would find that the entire system SAE

behaves as a pure state |ψ〉 which may be a ldquocat staterdquo involving the superposition

of macroscopically different matter configurations Decoherence thus arises from the

description of the world by an observer who has access only to a subsystem To the

extent that the environment is defined by what a given observer cannot measure in

practice decoherence is subjective

The subjectivity of decoherence is not a problem as long as we are content to

explain our own experience ie that of an observer immersed in a much larger system

But the lack of any environment implies that decoherence cannot occur in a complete

unitary description of the whole universe It is possible that no such description exists

for our universe In Sec 21 we will argue however that causality places restrictions

on decoherence in much smaller regions in which the applicability of unitary quantum-

mechanical evolution seems beyond doubt

In Sec 22 we apply our analysis of decoherence and causality to eternal infla-

tion We will obtain a straightforward but perhaps surprising consequence in a global

description of an eternally inflating spacetime decoherence cannot occur so it is in-

consistent to imagine that pocket universes or vacuum bubbles nucleate at particular

locations and times In Sec 23 we discuss a number of attempts to rescue a unitary

global description and conclude that they do not succeed

In Sec 24 we review the ldquocausal diamondrdquo description of the multiverse The

causal diamond is the largest spacetime region that can be causally probed and it can

be thought of as the past light-cone from a point on the future conformal boundary

We argue that the causal diamond description leads to a natural observer-independent

choice of environment because its boundary is light-like it acts as a one-way membrane

and degrees of freedom that leave the diamond do not return except in very special

ndash 4 ndash

cases These degrees of freedom can be traced over leading to a branching tree of

causal diamond histories

Next we turn to the question of whether the global picture of the multiverse can

be recovered from the decoherent causal diamonds In Sec 25 we review a known

duality between the causal diamond and a particular foliation of the global geometry

known as light-cone time both give the same probabilities This duality took the

standard global picture as a starting point but in Sec 26 we reinterpret it as a way of

reconstructing the global viewpoint from the local one If the causal diamond histories

are the many-worlds this construction shows that the multiverse is the many-worlds

pieced together in a single geometry

In Sec 3 we turn to the second limitation associated with decoherence its re-

versibility Consider a causal diamond with finite maximal boundary area Amax En-

tropy bounds imply that such diamonds can be described by a Hilbert space with finite

dimension no greater than exp(Amax2) [67]3 This means that no observables in such

diamonds can be defined with infinite precision In Sec 31 and 32 we will discuss

another implication of this finiteness there is a tiny but nonzero probability that deco-

herence will be undone This means that the decoherent histories of causal diamonds

and the reconstruction of a global spacetime from such diamonds is not completely

exact

No matter how good an approximation is it is important to understand the precise

statement that it is an approximation to In Sec 33 we will develop two postulates

that should be satisfied by a fundamental quantum-mechanical theory if decoherence

is to be sharp and the associated probabilities operationally meaningful decoherence

must be irreversible and it must occur infinitely many times for a given experiment in

a single causally connected region

The string landscape contains supersymmetric vacua with exactly vanishing cosmo-

logical constant Causal diamonds which enter such vacua have infinite boundary area

at late times We argue in Sec 34 that in these ldquohatrdquo regions all our postulates can

be satisfied Exact observables can exist and decoherence by the mechanism of Sec 24

can be truly irreversible Moreover because the hat is a spatially open statistically

homogeneous universe anything that happens in the hat will happen infinitely many

times

In Sec 35 we review black hole complementarity and we conjecture an analogous

ldquohat complementarityrdquo for the multiverse It ensures that the approximate observables

and approximate decoherence of causal diamonds with finite area (Sec 24) have precise

counterparts in the hat In Sec 36 we propose a relation between the global multiverse

3This point has long been emphasized by Banks and Fischler [8ndash10]

ndash 5 ndash

M

E S+A

Figure 1 Decoherence and causality At the event M a macroscopic apparatus A becomes

correlated with a quantum system S Thereafter environmental degrees of freedom E interact

with the apparatus In practice an observer viewing the apparatus is ignorant of the exact

state of the environment and so must trace over this Hilbert space factor This results in a

mixed state which is diagonal in a particular ldquopointerrdquo basis picked out by the interaction

between E and A The state of the full system SAE however remains pure In particular

decoherence does not take place and no preferred bases arises in a complete description of

any region larger than the future lightcone of M

reconstruction of Sec 26 and the Census Taker cutoff [11] on the hat geometry

Two interesting papers have recently explored relations between the many-worlds

interpretation and the multiverse [12 13] The present work differs substantially in a

number of aspects Among them is the notion that causal diamonds provide a pre-

ferred environment for decoherence our view of the global multiverse as a patchwork

of decoherent causal diamonds our postulates requiring irreversible entanglement and

infinite repetition and the associated role we ascribe to hat regions of the multiverse

2 Building the multiverse from the many worlds of causal di-

amonds

21 Decoherence and causality

The decoherence mechanism reviewed above relies on ignoring the degrees of freedom

that a given observer fails to monitor which is fine if our goal is to explain the ex-

ndash 6 ndash

periences of that observer But this subjective viewpoint clashes with the impersonal

unitary description of large spacetime regionsmdashthe viewpoint usually adopted in cos-

mology We are free of course to pick any subsystem and trace over it But the

outcome will depend on this choice The usual choices implicitly involve locality but

not in a unique way

For example we might choose S to be an electron and E to be the inanimate

laboratory The systemrsquos wave function collapses when the electron becomes entangled

with some detector But we may also include in S everything out to the edge of the

solar system The environment is whatever is out beyond the orbit of Pluto In that

case the collapse of the system wavefunction cannot take place until a photon from the

detector has passed Plutorsquos orbit This would take about a five hours during which the

system wavefunction is coherent

In particular decoherence cannot occur in the complete quantum description of

any region larger than the future light-cone of the measurement event M (Fig 1) All

environmental degrees of freedom that could have become entangled with the apparatus

since the measurement took place must lie within this lightcone and hence are included

not traced over in a complete description of the state An example of such a region

is the whole universe ie any Cauchy surface to the future of M But at least at

sufficiently early times the future light-cone of M will be much smaller than the whole

universe Already on this scale the system SAE will be coherent

In our earlier example suppose that we measure the spin of an electron that is

initially prepared in a superposition of spin-up and spin-down a|0〉S + b|1〉S resulting

in the state |ψ〉 of Eq (12) A complete description of the solar system (defined as

the interior of a sphere the size of Plutorsquos orbit with a light-crossing time of about

10 hours) by a local quantum field theory contains every particle that could possibly

have interacted with the apparatus after the measurement for about 5 hours This

description would maintain the coherence of the macroscopic superpositions implicit in

the state |ψ〉 such as apparatus-up with apparatus-down until the first photons that

are entangled with the apparatus leave the solar system

Of course a detailed knowledge of the quantum state in such large regions is

unavailable to a realistic observer (Indeed if the region is larger than a cosmological

event horizon then its quantum state is cannot be probed at all without violating

causality) Yet our theoretical description of matter fields in spacetime retains in

principle all degrees of freedom and full coherence of the quantum state In theoretical

cosmology this can lead to inconsistencies if we describe regions that are larger than the

future light-cones of events that we nevertheless treat as decohered We now consider

an important example

ndash 7 ndash

22 Failure to decohere A problem with the global multiverse

The above analysis undermines what we will call the ldquostandard global picturerdquo of an

eternally inflating spacetime Consider an effective potential containing at least one

inflating false vacuum ie a metastable de Sitter vacuum with decay rate much less

than one decay per Hubble volume and Hubble time We will also assume that there

is at least one terminal vacuum with nonpositive cosmological constant (The string

theory landscape is believed to have at least 10100primes of vacua of both types [14ndash17])

According to the standard description of eternal inflation an inflating vacuum nu-

cleates bubble-universes in a statistical manner similar to the way superheated water

nucleates bubbles of steam That process is described by classical stochastic production

of bubbles which occurs randomly but the randomness is classical The bubbles nucle-

ate at definite locations and coherent quantum mechanical interference plays no role

The conventional description of eternal inflation similarly based on classical stochastic

processes However this picture is not consistent with a complete quantum-mechanical

description of a global region of the multiverse

To explain why this is so consider the future domain of dependence D(Σ0) of a

sufficiently large hypersurface Σ0 which need not be a Cauchy surface D(Σ0) consists

of all events that can be predicted from data on Σ0 see Fig 2 If Σ0 contains suffi-

ciently large and long-lived metastable de Sitter regions then bubbles of vacua of lower

energy do not consume the parent de Sitter vacua in which they nucleate [18] Hence

the de Sitter vacua are said to inflate eternally producing an unbounded number of

bubble universes The resulting spacetime is said to have the structure shown in the

conformal diagram in Fig 2 with bubbles nucleating at definite spacetime events The

future conformal boundary is spacelike in regions with negative cosmological constant

corresponding to a local big crunch The boundary contains null ldquohatsrdquo in regions

occupied by vacua with Λ = 0

But this picture does not arise in a complete quantum description of D(Σ0) The

future light-cones of events at late times are much smaller than D(Σ0) In any state

that describes the entire spacetime region D(Σ0) decoherence can only take place at

the margin of D(Σ0) (shown light shaded in Fig 2) in the region from which particles

can escape into the complement of D(Σ0) in the full spacetime No decoherence can

take place in the infinite spacetime region defined by the past domain of dependence

of the future boundary of D(Σ0) In this region quantum evolution remains coherent

even if it results in the superposition of macroscopically distinct matter or spacetime

configurations

An important example is the superposition of vacuum decays taking place at dif-

ferent places Without decoherence it makes no sense to say that bubbles nucleate at

ndash 8 ndash

future boundary

Σ0

Figure 2 The future domain of dependence D(Σ0) (light or dark shaded) is the spacetime

region that can be predicted from data on the timeslice Σ0 If the future conformal boundary

contains spacelike portions as in eternal inflation or inside a black hole then the future

light-cones of events in the dark shaded region remain entirely within D(Σ0) Pure quantum

states do not decohere in this region in a complete description of D(Σ0) This is true even for

states that involve macroscopic superpositions such as the locations of pocket universes in

eternal inflation (dashed lines) calling into question the self-consistency of the global picture

of eternal inflation

particular times and locations rather a wavefunction with initial support only in the

parent vacuum develops into a superposition of parent and daughter vacua Bubbles

nucleating at all places at times are ldquoquantum superimposedrdquo With the gravitational

backreaction included the metric too would remain in a quantum-mechanical super-

position This contradicts the standard global picture of eternal inflation in which

domain walls vacua and the spacetime metric take on definite values as if drawn from

a density matrix obtained by tracing over some degrees of freedom and as if the inter-

action with these degrees of freedom had picked out a preferred basis that eliminates

the quantum superposition of bubbles and vacua

Let us quickly get rid of one red herring Can the standard geometry of eternal

inflation be recovered by using so-called semi-classical gravity in which the metric is

sourced by the expectation value of the energy-momentum tensor

Gmicroν = 8π〈Tmicroν〉 (21)

This does not work because the matter quantum fields would still remain coherent At

the level of the quantum fields the wavefunction initially has support only in the false

vacuum Over time it evolves to a superposition of the false vacuum (with decreasing

amplitude) with the true vacuum (with increasing amplitude) plus a superposition

of expanding and colliding domain walls This state is quite complicated but the

expectation value of its stress tensor should remain spatially homogeneous if it was so

initially The net effect over time would be a continuous conversion of vacuum energy

into ordinary matter or radiation (from the collision of bubbles and motion of the scalar

field) By Eq (21) the geometry spacetime would respond to the homogeneous glide

ndash 9 ndash

of the vacuum energy to negative values This would result in a global crunch after

finite time in stark contrast to the standard picture of global eternal inflation In

any case it seems implausible that semi-classical gravity should apply in a situation in

which coherent branches of the wavefunction have radically different gravitational back-

reaction The AdSCFT correspondence provides an explicit counterexample since the

superposition of two CFT states that correspond to different classical geometries must

correspond to a quantum superposition of the two metrics

The conclusion that we come to from these considerations is not that the global

multiverse is meaningless but that the parallel view should not be implemented by

unitary quantum mechanics But is there an alternative Can the standard global

picture be recovered by considering an observer who has access only to some of the

degrees of freedom of the multiverse and appealing to decoherence We debate this

question in the following section

23 Simpliciorsquos proposal

Simplicio and Sagredo have studied Sections 21 and 22 supplied to them by Salviati

They meet at Sagredorsquos house for a discussion

Simplicio You have convinced me that a complete description of eternal inflation

by unitary quantum evolution on global slices will not lead to a picture in which bubbles

form at definite places and times But all I need is an observer somewhere Then I

can take this observerrsquos point of view and trace over the degrees of freedom that are

inaccessible to him This decoheres events such as bubble nucleations in the entire

global multiverse It actually helps that some regions are causally disconnected from the

observer this makes his environmentmdashthe degrees of freedom he fails to accessmdashreally

huge

Sagredo An interesting idea But you seem to include everything outside the

observerrsquos horizon region in what you call the environment Once you trace over it it

is gone from your description and you could not possibly recover a global spacetime

Simplicio Your objection is valid but it also shows me how to fix my proposal

The observer should only trace over environmental degrees in his own horizon Deco-

herence is very efficient so this should suffice

Sagredo I wonder what would happen if there were two observers in widely

separated regions If one observerrsquos environment is enough to decohere the whole

universe which one should we pick

Simplicio I have not done a calculation but it seems to me that it shouldnrsquot

matter The outcome of an experiment by one of the observers should be the same no

matter which observerrsquos environment I trace over That is certainly how it works when

you and I both stare at the same apparatus

ndash 10 ndash

future boundary

Σ0

P O

Figure 3 Environmental degrees of freedom entangled with an observer at O remain within

the causal future of the causal past of O J+[Jminus(O)] (cyanshaded) They are not entangled

with distant regions of the multiverse Tracing over them will not lead to decoherence of

a bubble nucleated at P for example and hence will fail to reproduce the standard global

picture of eternal inflation

Sagredo Something is different about the multiverse When you and I both

observe Salviati we all become correlated by interactions with a common environment

But how does an observer in one horizon volume become correlated with an object in

another horizon volume far away

Salviati Sagredo you hit the nail on the head Decoherence requires the inter-

action of environmental degrees of freedom with the apparatus and the observer This

entangles them and it leads to a density matrix once the environment is ignored by the

observer But an observer cannot have interacted with degrees of freedom that were

never present in his past light-cone

Sagredo Thank you for articulating so clearly what to me was only a vague

concern Simplicio you look puzzled so let me summarize our objection in my own

words You proposed a method for obtaining the standard global picture of eternal

inflation you claim that we need only identify an arbitrary observer in the multiverse

and trace over his environment If we defined the environment as all degrees of freedom

the observer fails to monitor then it would include the causally disconnected regions

outside his horizon With this definition these regions will disappear entirely from your

description in conflict with the global picture So we agreed to define the environment

as the degrees of freedom that have interacted with the observer and which he cannot

access in practice But in this case the environment includes no degrees of freedom

outside the causal future of the observerrsquos causal past I have drawn this region in

Fig 3 But tracing over an environment can only decohere degrees of freedom that it

is entangled with In this case it can decohere some events that lie in the observerrsquos

past light-cone But it cannot affect quantum coherence in far-away horizon regions

because the environment you have picked is not entangled with these regions In those

ndash 11 ndash

regions bubble walls and vacua will remain in superposition which again conflicts with

the standard global picture of eternal inflation

Simplicio I see that my idea still has some problems I will need to identify more

than one observer-environment pair In fact if I wish to preserve the global picture

of the multiverse I will have to assume that an observer is present in every horizon

volume at all times Otherwise there will be horizon regions where no one is around

to decide which degrees of freedom are hard to keep track of so there is no way to

identify and trace over an environment In such regions bubbles would not form at

particular places and times in conflict with the standard global picture

Sagredo But this assumption is clearly violated in many landscape models Most

de Sitter vacua have large cosmological constant so that a single horizon volume is too

small to contain the large number of degrees of freedom required for an observer And

regions with small vacuum energy may be very long lived so the corresponding bubbles

contain many horizon volumes that are completely empty Irsquom afraid Simplicio that

your efforts to rescue the global multiverse are destined to fail

Salviati Why donrsquot we back up a little and return to Simpliciorsquos initial sug-

gestion Sagredo you objected that everything outside an observerrsquos horizon would

naturally be part of his environment and would be gone from our description if we

trace over it

Sagredo which means that the whole global description would be gone

Salviati but why is that a problem No observer inside the universe can ever

see more than what is in their past light-cone at late times or more precisely in their

causal diamond We may not be able to recover the global picture by tracing over

the region behind an observerrsquos horizon but the same procedure might well achieve

decoherence in the region the observer can actually access In fact we donrsquot even

need an actual observer we can get decoherence by tracing over degrees of freedom

that leave the causal horizon of any worldline This will allow us to say that a bubble

formed in one place and not another So why donrsquot we give up on the global description

for a moment Later on we can check whether a global picture can be recovered in

some way from the decoherent causal diamonds

Salviati hands out Sections 24ndash26

24 Objective decoherence from the causal diamond

If Hawking radiation contains the full information about the quantum state of a star

that collapsed to form a black hole then there is an apparent paradox The star is

located inside the black hole at spacelike separation from the Hawking cloud hence two

copies of the original quantum information are present simultaneously The xeroxing of

quantum information however conflicts with the linearity of quantum mechanics [19]

ndash 12 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

mechanical predictions We argue in Sec 2 that causal diamonds provide a natural

definition of environment in the multiverse leading to an observer-independent notion

of decoherent histories In Sec 3 we argue that these histories have precise irreversible

counterparts in the ldquohatrdquo-regions of the multiverse We now give a more detailed

overview of this paper

Outline In Sec 2 we address the first limitation of decoherence its subjectivity Be-

cause coherence is never lost in the full Hilbert space SAE the speed extent and

possible outcomes of decoherence depend on the definition of the environment E This

choice is made implicitly by an observer based on practical circumstances the envi-

ronment consists of degrees of freedom that have become entangled with the system

and apparatus but remain unobserved It is impractical for example to keep track of

every thermal photon emitted by a table of all of its interactions with light and air

particles and so on But if we did then we would find that the entire system SAE

behaves as a pure state |ψ〉 which may be a ldquocat staterdquo involving the superposition

of macroscopically different matter configurations Decoherence thus arises from the

description of the world by an observer who has access only to a subsystem To the

extent that the environment is defined by what a given observer cannot measure in

practice decoherence is subjective

The subjectivity of decoherence is not a problem as long as we are content to

explain our own experience ie that of an observer immersed in a much larger system

But the lack of any environment implies that decoherence cannot occur in a complete

unitary description of the whole universe It is possible that no such description exists

for our universe In Sec 21 we will argue however that causality places restrictions

on decoherence in much smaller regions in which the applicability of unitary quantum-

mechanical evolution seems beyond doubt

In Sec 22 we apply our analysis of decoherence and causality to eternal infla-

tion We will obtain a straightforward but perhaps surprising consequence in a global

description of an eternally inflating spacetime decoherence cannot occur so it is in-

consistent to imagine that pocket universes or vacuum bubbles nucleate at particular

locations and times In Sec 23 we discuss a number of attempts to rescue a unitary

global description and conclude that they do not succeed

In Sec 24 we review the ldquocausal diamondrdquo description of the multiverse The

causal diamond is the largest spacetime region that can be causally probed and it can

be thought of as the past light-cone from a point on the future conformal boundary

We argue that the causal diamond description leads to a natural observer-independent

choice of environment because its boundary is light-like it acts as a one-way membrane

and degrees of freedom that leave the diamond do not return except in very special

ndash 4 ndash

cases These degrees of freedom can be traced over leading to a branching tree of

causal diamond histories

Next we turn to the question of whether the global picture of the multiverse can

be recovered from the decoherent causal diamonds In Sec 25 we review a known

duality between the causal diamond and a particular foliation of the global geometry

known as light-cone time both give the same probabilities This duality took the

standard global picture as a starting point but in Sec 26 we reinterpret it as a way of

reconstructing the global viewpoint from the local one If the causal diamond histories

are the many-worlds this construction shows that the multiverse is the many-worlds

pieced together in a single geometry

In Sec 3 we turn to the second limitation associated with decoherence its re-

versibility Consider a causal diamond with finite maximal boundary area Amax En-

tropy bounds imply that such diamonds can be described by a Hilbert space with finite

dimension no greater than exp(Amax2) [67]3 This means that no observables in such

diamonds can be defined with infinite precision In Sec 31 and 32 we will discuss

another implication of this finiteness there is a tiny but nonzero probability that deco-

herence will be undone This means that the decoherent histories of causal diamonds

and the reconstruction of a global spacetime from such diamonds is not completely

exact

No matter how good an approximation is it is important to understand the precise

statement that it is an approximation to In Sec 33 we will develop two postulates

that should be satisfied by a fundamental quantum-mechanical theory if decoherence

is to be sharp and the associated probabilities operationally meaningful decoherence

must be irreversible and it must occur infinitely many times for a given experiment in

a single causally connected region

The string landscape contains supersymmetric vacua with exactly vanishing cosmo-

logical constant Causal diamonds which enter such vacua have infinite boundary area

at late times We argue in Sec 34 that in these ldquohatrdquo regions all our postulates can

be satisfied Exact observables can exist and decoherence by the mechanism of Sec 24

can be truly irreversible Moreover because the hat is a spatially open statistically

homogeneous universe anything that happens in the hat will happen infinitely many

times

In Sec 35 we review black hole complementarity and we conjecture an analogous

ldquohat complementarityrdquo for the multiverse It ensures that the approximate observables

and approximate decoherence of causal diamonds with finite area (Sec 24) have precise

counterparts in the hat In Sec 36 we propose a relation between the global multiverse

3This point has long been emphasized by Banks and Fischler [8ndash10]

ndash 5 ndash

M

E S+A

Figure 1 Decoherence and causality At the event M a macroscopic apparatus A becomes

correlated with a quantum system S Thereafter environmental degrees of freedom E interact

with the apparatus In practice an observer viewing the apparatus is ignorant of the exact

state of the environment and so must trace over this Hilbert space factor This results in a

mixed state which is diagonal in a particular ldquopointerrdquo basis picked out by the interaction

between E and A The state of the full system SAE however remains pure In particular

decoherence does not take place and no preferred bases arises in a complete description of

any region larger than the future lightcone of M

reconstruction of Sec 26 and the Census Taker cutoff [11] on the hat geometry

Two interesting papers have recently explored relations between the many-worlds

interpretation and the multiverse [12 13] The present work differs substantially in a

number of aspects Among them is the notion that causal diamonds provide a pre-

ferred environment for decoherence our view of the global multiverse as a patchwork

of decoherent causal diamonds our postulates requiring irreversible entanglement and

infinite repetition and the associated role we ascribe to hat regions of the multiverse

2 Building the multiverse from the many worlds of causal di-

amonds

21 Decoherence and causality

The decoherence mechanism reviewed above relies on ignoring the degrees of freedom

that a given observer fails to monitor which is fine if our goal is to explain the ex-

ndash 6 ndash

periences of that observer But this subjective viewpoint clashes with the impersonal

unitary description of large spacetime regionsmdashthe viewpoint usually adopted in cos-

mology We are free of course to pick any subsystem and trace over it But the

outcome will depend on this choice The usual choices implicitly involve locality but

not in a unique way

For example we might choose S to be an electron and E to be the inanimate

laboratory The systemrsquos wave function collapses when the electron becomes entangled

with some detector But we may also include in S everything out to the edge of the

solar system The environment is whatever is out beyond the orbit of Pluto In that

case the collapse of the system wavefunction cannot take place until a photon from the

detector has passed Plutorsquos orbit This would take about a five hours during which the

system wavefunction is coherent

In particular decoherence cannot occur in the complete quantum description of

any region larger than the future light-cone of the measurement event M (Fig 1) All

environmental degrees of freedom that could have become entangled with the apparatus

since the measurement took place must lie within this lightcone and hence are included

not traced over in a complete description of the state An example of such a region

is the whole universe ie any Cauchy surface to the future of M But at least at

sufficiently early times the future light-cone of M will be much smaller than the whole

universe Already on this scale the system SAE will be coherent

In our earlier example suppose that we measure the spin of an electron that is

initially prepared in a superposition of spin-up and spin-down a|0〉S + b|1〉S resulting

in the state |ψ〉 of Eq (12) A complete description of the solar system (defined as

the interior of a sphere the size of Plutorsquos orbit with a light-crossing time of about

10 hours) by a local quantum field theory contains every particle that could possibly

have interacted with the apparatus after the measurement for about 5 hours This

description would maintain the coherence of the macroscopic superpositions implicit in

the state |ψ〉 such as apparatus-up with apparatus-down until the first photons that

are entangled with the apparatus leave the solar system

Of course a detailed knowledge of the quantum state in such large regions is

unavailable to a realistic observer (Indeed if the region is larger than a cosmological

event horizon then its quantum state is cannot be probed at all without violating

causality) Yet our theoretical description of matter fields in spacetime retains in

principle all degrees of freedom and full coherence of the quantum state In theoretical

cosmology this can lead to inconsistencies if we describe regions that are larger than the

future light-cones of events that we nevertheless treat as decohered We now consider

an important example

ndash 7 ndash

22 Failure to decohere A problem with the global multiverse

The above analysis undermines what we will call the ldquostandard global picturerdquo of an

eternally inflating spacetime Consider an effective potential containing at least one

inflating false vacuum ie a metastable de Sitter vacuum with decay rate much less

than one decay per Hubble volume and Hubble time We will also assume that there

is at least one terminal vacuum with nonpositive cosmological constant (The string

theory landscape is believed to have at least 10100primes of vacua of both types [14ndash17])

According to the standard description of eternal inflation an inflating vacuum nu-

cleates bubble-universes in a statistical manner similar to the way superheated water

nucleates bubbles of steam That process is described by classical stochastic production

of bubbles which occurs randomly but the randomness is classical The bubbles nucle-

ate at definite locations and coherent quantum mechanical interference plays no role

The conventional description of eternal inflation similarly based on classical stochastic

processes However this picture is not consistent with a complete quantum-mechanical

description of a global region of the multiverse

To explain why this is so consider the future domain of dependence D(Σ0) of a

sufficiently large hypersurface Σ0 which need not be a Cauchy surface D(Σ0) consists

of all events that can be predicted from data on Σ0 see Fig 2 If Σ0 contains suffi-

ciently large and long-lived metastable de Sitter regions then bubbles of vacua of lower

energy do not consume the parent de Sitter vacua in which they nucleate [18] Hence

the de Sitter vacua are said to inflate eternally producing an unbounded number of

bubble universes The resulting spacetime is said to have the structure shown in the

conformal diagram in Fig 2 with bubbles nucleating at definite spacetime events The

future conformal boundary is spacelike in regions with negative cosmological constant

corresponding to a local big crunch The boundary contains null ldquohatsrdquo in regions

occupied by vacua with Λ = 0

But this picture does not arise in a complete quantum description of D(Σ0) The

future light-cones of events at late times are much smaller than D(Σ0) In any state

that describes the entire spacetime region D(Σ0) decoherence can only take place at

the margin of D(Σ0) (shown light shaded in Fig 2) in the region from which particles

can escape into the complement of D(Σ0) in the full spacetime No decoherence can

take place in the infinite spacetime region defined by the past domain of dependence

of the future boundary of D(Σ0) In this region quantum evolution remains coherent

even if it results in the superposition of macroscopically distinct matter or spacetime

configurations

An important example is the superposition of vacuum decays taking place at dif-

ferent places Without decoherence it makes no sense to say that bubbles nucleate at

ndash 8 ndash

future boundary

Σ0

Figure 2 The future domain of dependence D(Σ0) (light or dark shaded) is the spacetime

region that can be predicted from data on the timeslice Σ0 If the future conformal boundary

contains spacelike portions as in eternal inflation or inside a black hole then the future

light-cones of events in the dark shaded region remain entirely within D(Σ0) Pure quantum

states do not decohere in this region in a complete description of D(Σ0) This is true even for

states that involve macroscopic superpositions such as the locations of pocket universes in

eternal inflation (dashed lines) calling into question the self-consistency of the global picture

of eternal inflation

particular times and locations rather a wavefunction with initial support only in the

parent vacuum develops into a superposition of parent and daughter vacua Bubbles

nucleating at all places at times are ldquoquantum superimposedrdquo With the gravitational

backreaction included the metric too would remain in a quantum-mechanical super-

position This contradicts the standard global picture of eternal inflation in which

domain walls vacua and the spacetime metric take on definite values as if drawn from

a density matrix obtained by tracing over some degrees of freedom and as if the inter-

action with these degrees of freedom had picked out a preferred basis that eliminates

the quantum superposition of bubbles and vacua

Let us quickly get rid of one red herring Can the standard geometry of eternal

inflation be recovered by using so-called semi-classical gravity in which the metric is

sourced by the expectation value of the energy-momentum tensor

Gmicroν = 8π〈Tmicroν〉 (21)

This does not work because the matter quantum fields would still remain coherent At

the level of the quantum fields the wavefunction initially has support only in the false

vacuum Over time it evolves to a superposition of the false vacuum (with decreasing

amplitude) with the true vacuum (with increasing amplitude) plus a superposition

of expanding and colliding domain walls This state is quite complicated but the

expectation value of its stress tensor should remain spatially homogeneous if it was so

initially The net effect over time would be a continuous conversion of vacuum energy

into ordinary matter or radiation (from the collision of bubbles and motion of the scalar

field) By Eq (21) the geometry spacetime would respond to the homogeneous glide

ndash 9 ndash

of the vacuum energy to negative values This would result in a global crunch after

finite time in stark contrast to the standard picture of global eternal inflation In

any case it seems implausible that semi-classical gravity should apply in a situation in

which coherent branches of the wavefunction have radically different gravitational back-

reaction The AdSCFT correspondence provides an explicit counterexample since the

superposition of two CFT states that correspond to different classical geometries must

correspond to a quantum superposition of the two metrics

The conclusion that we come to from these considerations is not that the global

multiverse is meaningless but that the parallel view should not be implemented by

unitary quantum mechanics But is there an alternative Can the standard global

picture be recovered by considering an observer who has access only to some of the

degrees of freedom of the multiverse and appealing to decoherence We debate this

question in the following section

23 Simpliciorsquos proposal

Simplicio and Sagredo have studied Sections 21 and 22 supplied to them by Salviati

They meet at Sagredorsquos house for a discussion

Simplicio You have convinced me that a complete description of eternal inflation

by unitary quantum evolution on global slices will not lead to a picture in which bubbles

form at definite places and times But all I need is an observer somewhere Then I

can take this observerrsquos point of view and trace over the degrees of freedom that are

inaccessible to him This decoheres events such as bubble nucleations in the entire

global multiverse It actually helps that some regions are causally disconnected from the

observer this makes his environmentmdashthe degrees of freedom he fails to accessmdashreally

huge

Sagredo An interesting idea But you seem to include everything outside the

observerrsquos horizon region in what you call the environment Once you trace over it it

is gone from your description and you could not possibly recover a global spacetime

Simplicio Your objection is valid but it also shows me how to fix my proposal

The observer should only trace over environmental degrees in his own horizon Deco-

herence is very efficient so this should suffice

Sagredo I wonder what would happen if there were two observers in widely

separated regions If one observerrsquos environment is enough to decohere the whole

universe which one should we pick

Simplicio I have not done a calculation but it seems to me that it shouldnrsquot

matter The outcome of an experiment by one of the observers should be the same no

matter which observerrsquos environment I trace over That is certainly how it works when

you and I both stare at the same apparatus

ndash 10 ndash

future boundary

Σ0

P O

Figure 3 Environmental degrees of freedom entangled with an observer at O remain within

the causal future of the causal past of O J+[Jminus(O)] (cyanshaded) They are not entangled

with distant regions of the multiverse Tracing over them will not lead to decoherence of

a bubble nucleated at P for example and hence will fail to reproduce the standard global

picture of eternal inflation

Sagredo Something is different about the multiverse When you and I both

observe Salviati we all become correlated by interactions with a common environment

But how does an observer in one horizon volume become correlated with an object in

another horizon volume far away

Salviati Sagredo you hit the nail on the head Decoherence requires the inter-

action of environmental degrees of freedom with the apparatus and the observer This

entangles them and it leads to a density matrix once the environment is ignored by the

observer But an observer cannot have interacted with degrees of freedom that were

never present in his past light-cone

Sagredo Thank you for articulating so clearly what to me was only a vague

concern Simplicio you look puzzled so let me summarize our objection in my own

words You proposed a method for obtaining the standard global picture of eternal

inflation you claim that we need only identify an arbitrary observer in the multiverse

and trace over his environment If we defined the environment as all degrees of freedom

the observer fails to monitor then it would include the causally disconnected regions

outside his horizon With this definition these regions will disappear entirely from your

description in conflict with the global picture So we agreed to define the environment

as the degrees of freedom that have interacted with the observer and which he cannot

access in practice But in this case the environment includes no degrees of freedom

outside the causal future of the observerrsquos causal past I have drawn this region in

Fig 3 But tracing over an environment can only decohere degrees of freedom that it

is entangled with In this case it can decohere some events that lie in the observerrsquos

past light-cone But it cannot affect quantum coherence in far-away horizon regions

because the environment you have picked is not entangled with these regions In those

ndash 11 ndash

regions bubble walls and vacua will remain in superposition which again conflicts with

the standard global picture of eternal inflation

Simplicio I see that my idea still has some problems I will need to identify more

than one observer-environment pair In fact if I wish to preserve the global picture

of the multiverse I will have to assume that an observer is present in every horizon

volume at all times Otherwise there will be horizon regions where no one is around

to decide which degrees of freedom are hard to keep track of so there is no way to

identify and trace over an environment In such regions bubbles would not form at

particular places and times in conflict with the standard global picture

Sagredo But this assumption is clearly violated in many landscape models Most

de Sitter vacua have large cosmological constant so that a single horizon volume is too

small to contain the large number of degrees of freedom required for an observer And

regions with small vacuum energy may be very long lived so the corresponding bubbles

contain many horizon volumes that are completely empty Irsquom afraid Simplicio that

your efforts to rescue the global multiverse are destined to fail

Salviati Why donrsquot we back up a little and return to Simpliciorsquos initial sug-

gestion Sagredo you objected that everything outside an observerrsquos horizon would

naturally be part of his environment and would be gone from our description if we

trace over it

Sagredo which means that the whole global description would be gone

Salviati but why is that a problem No observer inside the universe can ever

see more than what is in their past light-cone at late times or more precisely in their

causal diamond We may not be able to recover the global picture by tracing over

the region behind an observerrsquos horizon but the same procedure might well achieve

decoherence in the region the observer can actually access In fact we donrsquot even

need an actual observer we can get decoherence by tracing over degrees of freedom

that leave the causal horizon of any worldline This will allow us to say that a bubble

formed in one place and not another So why donrsquot we give up on the global description

for a moment Later on we can check whether a global picture can be recovered in

some way from the decoherent causal diamonds

Salviati hands out Sections 24ndash26

24 Objective decoherence from the causal diamond

If Hawking radiation contains the full information about the quantum state of a star

that collapsed to form a black hole then there is an apparent paradox The star is

located inside the black hole at spacelike separation from the Hawking cloud hence two

copies of the original quantum information are present simultaneously The xeroxing of

quantum information however conflicts with the linearity of quantum mechanics [19]

ndash 12 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

cases These degrees of freedom can be traced over leading to a branching tree of

causal diamond histories

Next we turn to the question of whether the global picture of the multiverse can

be recovered from the decoherent causal diamonds In Sec 25 we review a known

duality between the causal diamond and a particular foliation of the global geometry

known as light-cone time both give the same probabilities This duality took the

standard global picture as a starting point but in Sec 26 we reinterpret it as a way of

reconstructing the global viewpoint from the local one If the causal diamond histories

are the many-worlds this construction shows that the multiverse is the many-worlds

pieced together in a single geometry

In Sec 3 we turn to the second limitation associated with decoherence its re-

versibility Consider a causal diamond with finite maximal boundary area Amax En-

tropy bounds imply that such diamonds can be described by a Hilbert space with finite

dimension no greater than exp(Amax2) [67]3 This means that no observables in such

diamonds can be defined with infinite precision In Sec 31 and 32 we will discuss

another implication of this finiteness there is a tiny but nonzero probability that deco-

herence will be undone This means that the decoherent histories of causal diamonds

and the reconstruction of a global spacetime from such diamonds is not completely

exact

No matter how good an approximation is it is important to understand the precise

statement that it is an approximation to In Sec 33 we will develop two postulates

that should be satisfied by a fundamental quantum-mechanical theory if decoherence

is to be sharp and the associated probabilities operationally meaningful decoherence

must be irreversible and it must occur infinitely many times for a given experiment in

a single causally connected region

The string landscape contains supersymmetric vacua with exactly vanishing cosmo-

logical constant Causal diamonds which enter such vacua have infinite boundary area

at late times We argue in Sec 34 that in these ldquohatrdquo regions all our postulates can

be satisfied Exact observables can exist and decoherence by the mechanism of Sec 24

can be truly irreversible Moreover because the hat is a spatially open statistically

homogeneous universe anything that happens in the hat will happen infinitely many

times

In Sec 35 we review black hole complementarity and we conjecture an analogous

ldquohat complementarityrdquo for the multiverse It ensures that the approximate observables

and approximate decoherence of causal diamonds with finite area (Sec 24) have precise

counterparts in the hat In Sec 36 we propose a relation between the global multiverse

3This point has long been emphasized by Banks and Fischler [8ndash10]

ndash 5 ndash

M

E S+A

Figure 1 Decoherence and causality At the event M a macroscopic apparatus A becomes

correlated with a quantum system S Thereafter environmental degrees of freedom E interact

with the apparatus In practice an observer viewing the apparatus is ignorant of the exact

state of the environment and so must trace over this Hilbert space factor This results in a

mixed state which is diagonal in a particular ldquopointerrdquo basis picked out by the interaction

between E and A The state of the full system SAE however remains pure In particular

decoherence does not take place and no preferred bases arises in a complete description of

any region larger than the future lightcone of M

reconstruction of Sec 26 and the Census Taker cutoff [11] on the hat geometry

Two interesting papers have recently explored relations between the many-worlds

interpretation and the multiverse [12 13] The present work differs substantially in a

number of aspects Among them is the notion that causal diamonds provide a pre-

ferred environment for decoherence our view of the global multiverse as a patchwork

of decoherent causal diamonds our postulates requiring irreversible entanglement and

infinite repetition and the associated role we ascribe to hat regions of the multiverse

2 Building the multiverse from the many worlds of causal di-

amonds

21 Decoherence and causality

The decoherence mechanism reviewed above relies on ignoring the degrees of freedom

that a given observer fails to monitor which is fine if our goal is to explain the ex-

ndash 6 ndash

periences of that observer But this subjective viewpoint clashes with the impersonal

unitary description of large spacetime regionsmdashthe viewpoint usually adopted in cos-

mology We are free of course to pick any subsystem and trace over it But the

outcome will depend on this choice The usual choices implicitly involve locality but

not in a unique way

For example we might choose S to be an electron and E to be the inanimate

laboratory The systemrsquos wave function collapses when the electron becomes entangled

with some detector But we may also include in S everything out to the edge of the

solar system The environment is whatever is out beyond the orbit of Pluto In that

case the collapse of the system wavefunction cannot take place until a photon from the

detector has passed Plutorsquos orbit This would take about a five hours during which the

system wavefunction is coherent

In particular decoherence cannot occur in the complete quantum description of

any region larger than the future light-cone of the measurement event M (Fig 1) All

environmental degrees of freedom that could have become entangled with the apparatus

since the measurement took place must lie within this lightcone and hence are included

not traced over in a complete description of the state An example of such a region

is the whole universe ie any Cauchy surface to the future of M But at least at

sufficiently early times the future light-cone of M will be much smaller than the whole

universe Already on this scale the system SAE will be coherent

In our earlier example suppose that we measure the spin of an electron that is

initially prepared in a superposition of spin-up and spin-down a|0〉S + b|1〉S resulting

in the state |ψ〉 of Eq (12) A complete description of the solar system (defined as

the interior of a sphere the size of Plutorsquos orbit with a light-crossing time of about

10 hours) by a local quantum field theory contains every particle that could possibly

have interacted with the apparatus after the measurement for about 5 hours This

description would maintain the coherence of the macroscopic superpositions implicit in

the state |ψ〉 such as apparatus-up with apparatus-down until the first photons that

are entangled with the apparatus leave the solar system

Of course a detailed knowledge of the quantum state in such large regions is

unavailable to a realistic observer (Indeed if the region is larger than a cosmological

event horizon then its quantum state is cannot be probed at all without violating

causality) Yet our theoretical description of matter fields in spacetime retains in

principle all degrees of freedom and full coherence of the quantum state In theoretical

cosmology this can lead to inconsistencies if we describe regions that are larger than the

future light-cones of events that we nevertheless treat as decohered We now consider

an important example

ndash 7 ndash

22 Failure to decohere A problem with the global multiverse

The above analysis undermines what we will call the ldquostandard global picturerdquo of an

eternally inflating spacetime Consider an effective potential containing at least one

inflating false vacuum ie a metastable de Sitter vacuum with decay rate much less

than one decay per Hubble volume and Hubble time We will also assume that there

is at least one terminal vacuum with nonpositive cosmological constant (The string

theory landscape is believed to have at least 10100primes of vacua of both types [14ndash17])

According to the standard description of eternal inflation an inflating vacuum nu-

cleates bubble-universes in a statistical manner similar to the way superheated water

nucleates bubbles of steam That process is described by classical stochastic production

of bubbles which occurs randomly but the randomness is classical The bubbles nucle-

ate at definite locations and coherent quantum mechanical interference plays no role

The conventional description of eternal inflation similarly based on classical stochastic

processes However this picture is not consistent with a complete quantum-mechanical

description of a global region of the multiverse

To explain why this is so consider the future domain of dependence D(Σ0) of a

sufficiently large hypersurface Σ0 which need not be a Cauchy surface D(Σ0) consists

of all events that can be predicted from data on Σ0 see Fig 2 If Σ0 contains suffi-

ciently large and long-lived metastable de Sitter regions then bubbles of vacua of lower

energy do not consume the parent de Sitter vacua in which they nucleate [18] Hence

the de Sitter vacua are said to inflate eternally producing an unbounded number of

bubble universes The resulting spacetime is said to have the structure shown in the

conformal diagram in Fig 2 with bubbles nucleating at definite spacetime events The

future conformal boundary is spacelike in regions with negative cosmological constant

corresponding to a local big crunch The boundary contains null ldquohatsrdquo in regions

occupied by vacua with Λ = 0

But this picture does not arise in a complete quantum description of D(Σ0) The

future light-cones of events at late times are much smaller than D(Σ0) In any state

that describes the entire spacetime region D(Σ0) decoherence can only take place at

the margin of D(Σ0) (shown light shaded in Fig 2) in the region from which particles

can escape into the complement of D(Σ0) in the full spacetime No decoherence can

take place in the infinite spacetime region defined by the past domain of dependence

of the future boundary of D(Σ0) In this region quantum evolution remains coherent

even if it results in the superposition of macroscopically distinct matter or spacetime

configurations

An important example is the superposition of vacuum decays taking place at dif-

ferent places Without decoherence it makes no sense to say that bubbles nucleate at

ndash 8 ndash

future boundary

Σ0

Figure 2 The future domain of dependence D(Σ0) (light or dark shaded) is the spacetime

region that can be predicted from data on the timeslice Σ0 If the future conformal boundary

contains spacelike portions as in eternal inflation or inside a black hole then the future

light-cones of events in the dark shaded region remain entirely within D(Σ0) Pure quantum

states do not decohere in this region in a complete description of D(Σ0) This is true even for

states that involve macroscopic superpositions such as the locations of pocket universes in

eternal inflation (dashed lines) calling into question the self-consistency of the global picture

of eternal inflation

particular times and locations rather a wavefunction with initial support only in the

parent vacuum develops into a superposition of parent and daughter vacua Bubbles

nucleating at all places at times are ldquoquantum superimposedrdquo With the gravitational

backreaction included the metric too would remain in a quantum-mechanical super-

position This contradicts the standard global picture of eternal inflation in which

domain walls vacua and the spacetime metric take on definite values as if drawn from

a density matrix obtained by tracing over some degrees of freedom and as if the inter-

action with these degrees of freedom had picked out a preferred basis that eliminates

the quantum superposition of bubbles and vacua

Let us quickly get rid of one red herring Can the standard geometry of eternal

inflation be recovered by using so-called semi-classical gravity in which the metric is

sourced by the expectation value of the energy-momentum tensor

Gmicroν = 8π〈Tmicroν〉 (21)

This does not work because the matter quantum fields would still remain coherent At

the level of the quantum fields the wavefunction initially has support only in the false

vacuum Over time it evolves to a superposition of the false vacuum (with decreasing

amplitude) with the true vacuum (with increasing amplitude) plus a superposition

of expanding and colliding domain walls This state is quite complicated but the

expectation value of its stress tensor should remain spatially homogeneous if it was so

initially The net effect over time would be a continuous conversion of vacuum energy

into ordinary matter or radiation (from the collision of bubbles and motion of the scalar

field) By Eq (21) the geometry spacetime would respond to the homogeneous glide

ndash 9 ndash

of the vacuum energy to negative values This would result in a global crunch after

finite time in stark contrast to the standard picture of global eternal inflation In

any case it seems implausible that semi-classical gravity should apply in a situation in

which coherent branches of the wavefunction have radically different gravitational back-

reaction The AdSCFT correspondence provides an explicit counterexample since the

superposition of two CFT states that correspond to different classical geometries must

correspond to a quantum superposition of the two metrics

The conclusion that we come to from these considerations is not that the global

multiverse is meaningless but that the parallel view should not be implemented by

unitary quantum mechanics But is there an alternative Can the standard global

picture be recovered by considering an observer who has access only to some of the

degrees of freedom of the multiverse and appealing to decoherence We debate this

question in the following section

23 Simpliciorsquos proposal

Simplicio and Sagredo have studied Sections 21 and 22 supplied to them by Salviati

They meet at Sagredorsquos house for a discussion

Simplicio You have convinced me that a complete description of eternal inflation

by unitary quantum evolution on global slices will not lead to a picture in which bubbles

form at definite places and times But all I need is an observer somewhere Then I

can take this observerrsquos point of view and trace over the degrees of freedom that are

inaccessible to him This decoheres events such as bubble nucleations in the entire

global multiverse It actually helps that some regions are causally disconnected from the

observer this makes his environmentmdashthe degrees of freedom he fails to accessmdashreally

huge

Sagredo An interesting idea But you seem to include everything outside the

observerrsquos horizon region in what you call the environment Once you trace over it it

is gone from your description and you could not possibly recover a global spacetime

Simplicio Your objection is valid but it also shows me how to fix my proposal

The observer should only trace over environmental degrees in his own horizon Deco-

herence is very efficient so this should suffice

Sagredo I wonder what would happen if there were two observers in widely

separated regions If one observerrsquos environment is enough to decohere the whole

universe which one should we pick

Simplicio I have not done a calculation but it seems to me that it shouldnrsquot

matter The outcome of an experiment by one of the observers should be the same no

matter which observerrsquos environment I trace over That is certainly how it works when

you and I both stare at the same apparatus

ndash 10 ndash

future boundary

Σ0

P O

Figure 3 Environmental degrees of freedom entangled with an observer at O remain within

the causal future of the causal past of O J+[Jminus(O)] (cyanshaded) They are not entangled

with distant regions of the multiverse Tracing over them will not lead to decoherence of

a bubble nucleated at P for example and hence will fail to reproduce the standard global

picture of eternal inflation

Sagredo Something is different about the multiverse When you and I both

observe Salviati we all become correlated by interactions with a common environment

But how does an observer in one horizon volume become correlated with an object in

another horizon volume far away

Salviati Sagredo you hit the nail on the head Decoherence requires the inter-

action of environmental degrees of freedom with the apparatus and the observer This

entangles them and it leads to a density matrix once the environment is ignored by the

observer But an observer cannot have interacted with degrees of freedom that were

never present in his past light-cone

Sagredo Thank you for articulating so clearly what to me was only a vague

concern Simplicio you look puzzled so let me summarize our objection in my own

words You proposed a method for obtaining the standard global picture of eternal

inflation you claim that we need only identify an arbitrary observer in the multiverse

and trace over his environment If we defined the environment as all degrees of freedom

the observer fails to monitor then it would include the causally disconnected regions

outside his horizon With this definition these regions will disappear entirely from your

description in conflict with the global picture So we agreed to define the environment

as the degrees of freedom that have interacted with the observer and which he cannot

access in practice But in this case the environment includes no degrees of freedom

outside the causal future of the observerrsquos causal past I have drawn this region in

Fig 3 But tracing over an environment can only decohere degrees of freedom that it

is entangled with In this case it can decohere some events that lie in the observerrsquos

past light-cone But it cannot affect quantum coherence in far-away horizon regions

because the environment you have picked is not entangled with these regions In those

ndash 11 ndash

regions bubble walls and vacua will remain in superposition which again conflicts with

the standard global picture of eternal inflation

Simplicio I see that my idea still has some problems I will need to identify more

than one observer-environment pair In fact if I wish to preserve the global picture

of the multiverse I will have to assume that an observer is present in every horizon

volume at all times Otherwise there will be horizon regions where no one is around

to decide which degrees of freedom are hard to keep track of so there is no way to

identify and trace over an environment In such regions bubbles would not form at

particular places and times in conflict with the standard global picture

Sagredo But this assumption is clearly violated in many landscape models Most

de Sitter vacua have large cosmological constant so that a single horizon volume is too

small to contain the large number of degrees of freedom required for an observer And

regions with small vacuum energy may be very long lived so the corresponding bubbles

contain many horizon volumes that are completely empty Irsquom afraid Simplicio that

your efforts to rescue the global multiverse are destined to fail

Salviati Why donrsquot we back up a little and return to Simpliciorsquos initial sug-

gestion Sagredo you objected that everything outside an observerrsquos horizon would

naturally be part of his environment and would be gone from our description if we

trace over it

Sagredo which means that the whole global description would be gone

Salviati but why is that a problem No observer inside the universe can ever

see more than what is in their past light-cone at late times or more precisely in their

causal diamond We may not be able to recover the global picture by tracing over

the region behind an observerrsquos horizon but the same procedure might well achieve

decoherence in the region the observer can actually access In fact we donrsquot even

need an actual observer we can get decoherence by tracing over degrees of freedom

that leave the causal horizon of any worldline This will allow us to say that a bubble

formed in one place and not another So why donrsquot we give up on the global description

for a moment Later on we can check whether a global picture can be recovered in

some way from the decoherent causal diamonds

Salviati hands out Sections 24ndash26

24 Objective decoherence from the causal diamond

If Hawking radiation contains the full information about the quantum state of a star

that collapsed to form a black hole then there is an apparent paradox The star is

located inside the black hole at spacelike separation from the Hawking cloud hence two

copies of the original quantum information are present simultaneously The xeroxing of

quantum information however conflicts with the linearity of quantum mechanics [19]

ndash 12 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

M

E S+A

Figure 1 Decoherence and causality At the event M a macroscopic apparatus A becomes

correlated with a quantum system S Thereafter environmental degrees of freedom E interact

with the apparatus In practice an observer viewing the apparatus is ignorant of the exact

state of the environment and so must trace over this Hilbert space factor This results in a

mixed state which is diagonal in a particular ldquopointerrdquo basis picked out by the interaction

between E and A The state of the full system SAE however remains pure In particular

decoherence does not take place and no preferred bases arises in a complete description of

any region larger than the future lightcone of M

reconstruction of Sec 26 and the Census Taker cutoff [11] on the hat geometry

Two interesting papers have recently explored relations between the many-worlds

interpretation and the multiverse [12 13] The present work differs substantially in a

number of aspects Among them is the notion that causal diamonds provide a pre-

ferred environment for decoherence our view of the global multiverse as a patchwork

of decoherent causal diamonds our postulates requiring irreversible entanglement and

infinite repetition and the associated role we ascribe to hat regions of the multiverse

2 Building the multiverse from the many worlds of causal di-

amonds

21 Decoherence and causality

The decoherence mechanism reviewed above relies on ignoring the degrees of freedom

that a given observer fails to monitor which is fine if our goal is to explain the ex-

ndash 6 ndash

periences of that observer But this subjective viewpoint clashes with the impersonal

unitary description of large spacetime regionsmdashthe viewpoint usually adopted in cos-

mology We are free of course to pick any subsystem and trace over it But the

outcome will depend on this choice The usual choices implicitly involve locality but

not in a unique way

For example we might choose S to be an electron and E to be the inanimate

laboratory The systemrsquos wave function collapses when the electron becomes entangled

with some detector But we may also include in S everything out to the edge of the

solar system The environment is whatever is out beyond the orbit of Pluto In that

case the collapse of the system wavefunction cannot take place until a photon from the

detector has passed Plutorsquos orbit This would take about a five hours during which the

system wavefunction is coherent

In particular decoherence cannot occur in the complete quantum description of

any region larger than the future light-cone of the measurement event M (Fig 1) All

environmental degrees of freedom that could have become entangled with the apparatus

since the measurement took place must lie within this lightcone and hence are included

not traced over in a complete description of the state An example of such a region

is the whole universe ie any Cauchy surface to the future of M But at least at

sufficiently early times the future light-cone of M will be much smaller than the whole

universe Already on this scale the system SAE will be coherent

In our earlier example suppose that we measure the spin of an electron that is

initially prepared in a superposition of spin-up and spin-down a|0〉S + b|1〉S resulting

in the state |ψ〉 of Eq (12) A complete description of the solar system (defined as

the interior of a sphere the size of Plutorsquos orbit with a light-crossing time of about

10 hours) by a local quantum field theory contains every particle that could possibly

have interacted with the apparatus after the measurement for about 5 hours This

description would maintain the coherence of the macroscopic superpositions implicit in

the state |ψ〉 such as apparatus-up with apparatus-down until the first photons that

are entangled with the apparatus leave the solar system

Of course a detailed knowledge of the quantum state in such large regions is

unavailable to a realistic observer (Indeed if the region is larger than a cosmological

event horizon then its quantum state is cannot be probed at all without violating

causality) Yet our theoretical description of matter fields in spacetime retains in

principle all degrees of freedom and full coherence of the quantum state In theoretical

cosmology this can lead to inconsistencies if we describe regions that are larger than the

future light-cones of events that we nevertheless treat as decohered We now consider

an important example

ndash 7 ndash

22 Failure to decohere A problem with the global multiverse

The above analysis undermines what we will call the ldquostandard global picturerdquo of an

eternally inflating spacetime Consider an effective potential containing at least one

inflating false vacuum ie a metastable de Sitter vacuum with decay rate much less

than one decay per Hubble volume and Hubble time We will also assume that there

is at least one terminal vacuum with nonpositive cosmological constant (The string

theory landscape is believed to have at least 10100primes of vacua of both types [14ndash17])

According to the standard description of eternal inflation an inflating vacuum nu-

cleates bubble-universes in a statistical manner similar to the way superheated water

nucleates bubbles of steam That process is described by classical stochastic production

of bubbles which occurs randomly but the randomness is classical The bubbles nucle-

ate at definite locations and coherent quantum mechanical interference plays no role

The conventional description of eternal inflation similarly based on classical stochastic

processes However this picture is not consistent with a complete quantum-mechanical

description of a global region of the multiverse

To explain why this is so consider the future domain of dependence D(Σ0) of a

sufficiently large hypersurface Σ0 which need not be a Cauchy surface D(Σ0) consists

of all events that can be predicted from data on Σ0 see Fig 2 If Σ0 contains suffi-

ciently large and long-lived metastable de Sitter regions then bubbles of vacua of lower

energy do not consume the parent de Sitter vacua in which they nucleate [18] Hence

the de Sitter vacua are said to inflate eternally producing an unbounded number of

bubble universes The resulting spacetime is said to have the structure shown in the

conformal diagram in Fig 2 with bubbles nucleating at definite spacetime events The

future conformal boundary is spacelike in regions with negative cosmological constant

corresponding to a local big crunch The boundary contains null ldquohatsrdquo in regions

occupied by vacua with Λ = 0

But this picture does not arise in a complete quantum description of D(Σ0) The

future light-cones of events at late times are much smaller than D(Σ0) In any state

that describes the entire spacetime region D(Σ0) decoherence can only take place at

the margin of D(Σ0) (shown light shaded in Fig 2) in the region from which particles

can escape into the complement of D(Σ0) in the full spacetime No decoherence can

take place in the infinite spacetime region defined by the past domain of dependence

of the future boundary of D(Σ0) In this region quantum evolution remains coherent

even if it results in the superposition of macroscopically distinct matter or spacetime

configurations

An important example is the superposition of vacuum decays taking place at dif-

ferent places Without decoherence it makes no sense to say that bubbles nucleate at

ndash 8 ndash

future boundary

Σ0

Figure 2 The future domain of dependence D(Σ0) (light or dark shaded) is the spacetime

region that can be predicted from data on the timeslice Σ0 If the future conformal boundary

contains spacelike portions as in eternal inflation or inside a black hole then the future

light-cones of events in the dark shaded region remain entirely within D(Σ0) Pure quantum

states do not decohere in this region in a complete description of D(Σ0) This is true even for

states that involve macroscopic superpositions such as the locations of pocket universes in

eternal inflation (dashed lines) calling into question the self-consistency of the global picture

of eternal inflation

particular times and locations rather a wavefunction with initial support only in the

parent vacuum develops into a superposition of parent and daughter vacua Bubbles

nucleating at all places at times are ldquoquantum superimposedrdquo With the gravitational

backreaction included the metric too would remain in a quantum-mechanical super-

position This contradicts the standard global picture of eternal inflation in which

domain walls vacua and the spacetime metric take on definite values as if drawn from

a density matrix obtained by tracing over some degrees of freedom and as if the inter-

action with these degrees of freedom had picked out a preferred basis that eliminates

the quantum superposition of bubbles and vacua

Let us quickly get rid of one red herring Can the standard geometry of eternal

inflation be recovered by using so-called semi-classical gravity in which the metric is

sourced by the expectation value of the energy-momentum tensor

Gmicroν = 8π〈Tmicroν〉 (21)

This does not work because the matter quantum fields would still remain coherent At

the level of the quantum fields the wavefunction initially has support only in the false

vacuum Over time it evolves to a superposition of the false vacuum (with decreasing

amplitude) with the true vacuum (with increasing amplitude) plus a superposition

of expanding and colliding domain walls This state is quite complicated but the

expectation value of its stress tensor should remain spatially homogeneous if it was so

initially The net effect over time would be a continuous conversion of vacuum energy

into ordinary matter or radiation (from the collision of bubbles and motion of the scalar

field) By Eq (21) the geometry spacetime would respond to the homogeneous glide

ndash 9 ndash

of the vacuum energy to negative values This would result in a global crunch after

finite time in stark contrast to the standard picture of global eternal inflation In

any case it seems implausible that semi-classical gravity should apply in a situation in

which coherent branches of the wavefunction have radically different gravitational back-

reaction The AdSCFT correspondence provides an explicit counterexample since the

superposition of two CFT states that correspond to different classical geometries must

correspond to a quantum superposition of the two metrics

The conclusion that we come to from these considerations is not that the global

multiverse is meaningless but that the parallel view should not be implemented by

unitary quantum mechanics But is there an alternative Can the standard global

picture be recovered by considering an observer who has access only to some of the

degrees of freedom of the multiverse and appealing to decoherence We debate this

question in the following section

23 Simpliciorsquos proposal

Simplicio and Sagredo have studied Sections 21 and 22 supplied to them by Salviati

They meet at Sagredorsquos house for a discussion

Simplicio You have convinced me that a complete description of eternal inflation

by unitary quantum evolution on global slices will not lead to a picture in which bubbles

form at definite places and times But all I need is an observer somewhere Then I

can take this observerrsquos point of view and trace over the degrees of freedom that are

inaccessible to him This decoheres events such as bubble nucleations in the entire

global multiverse It actually helps that some regions are causally disconnected from the

observer this makes his environmentmdashthe degrees of freedom he fails to accessmdashreally

huge

Sagredo An interesting idea But you seem to include everything outside the

observerrsquos horizon region in what you call the environment Once you trace over it it

is gone from your description and you could not possibly recover a global spacetime

Simplicio Your objection is valid but it also shows me how to fix my proposal

The observer should only trace over environmental degrees in his own horizon Deco-

herence is very efficient so this should suffice

Sagredo I wonder what would happen if there were two observers in widely

separated regions If one observerrsquos environment is enough to decohere the whole

universe which one should we pick

Simplicio I have not done a calculation but it seems to me that it shouldnrsquot

matter The outcome of an experiment by one of the observers should be the same no

matter which observerrsquos environment I trace over That is certainly how it works when

you and I both stare at the same apparatus

ndash 10 ndash

future boundary

Σ0

P O

Figure 3 Environmental degrees of freedom entangled with an observer at O remain within

the causal future of the causal past of O J+[Jminus(O)] (cyanshaded) They are not entangled

with distant regions of the multiverse Tracing over them will not lead to decoherence of

a bubble nucleated at P for example and hence will fail to reproduce the standard global

picture of eternal inflation

Sagredo Something is different about the multiverse When you and I both

observe Salviati we all become correlated by interactions with a common environment

But how does an observer in one horizon volume become correlated with an object in

another horizon volume far away

Salviati Sagredo you hit the nail on the head Decoherence requires the inter-

action of environmental degrees of freedom with the apparatus and the observer This

entangles them and it leads to a density matrix once the environment is ignored by the

observer But an observer cannot have interacted with degrees of freedom that were

never present in his past light-cone

Sagredo Thank you for articulating so clearly what to me was only a vague

concern Simplicio you look puzzled so let me summarize our objection in my own

words You proposed a method for obtaining the standard global picture of eternal

inflation you claim that we need only identify an arbitrary observer in the multiverse

and trace over his environment If we defined the environment as all degrees of freedom

the observer fails to monitor then it would include the causally disconnected regions

outside his horizon With this definition these regions will disappear entirely from your

description in conflict with the global picture So we agreed to define the environment

as the degrees of freedom that have interacted with the observer and which he cannot

access in practice But in this case the environment includes no degrees of freedom

outside the causal future of the observerrsquos causal past I have drawn this region in

Fig 3 But tracing over an environment can only decohere degrees of freedom that it

is entangled with In this case it can decohere some events that lie in the observerrsquos

past light-cone But it cannot affect quantum coherence in far-away horizon regions

because the environment you have picked is not entangled with these regions In those

ndash 11 ndash

regions bubble walls and vacua will remain in superposition which again conflicts with

the standard global picture of eternal inflation

Simplicio I see that my idea still has some problems I will need to identify more

than one observer-environment pair In fact if I wish to preserve the global picture

of the multiverse I will have to assume that an observer is present in every horizon

volume at all times Otherwise there will be horizon regions where no one is around

to decide which degrees of freedom are hard to keep track of so there is no way to

identify and trace over an environment In such regions bubbles would not form at

particular places and times in conflict with the standard global picture

Sagredo But this assumption is clearly violated in many landscape models Most

de Sitter vacua have large cosmological constant so that a single horizon volume is too

small to contain the large number of degrees of freedom required for an observer And

regions with small vacuum energy may be very long lived so the corresponding bubbles

contain many horizon volumes that are completely empty Irsquom afraid Simplicio that

your efforts to rescue the global multiverse are destined to fail

Salviati Why donrsquot we back up a little and return to Simpliciorsquos initial sug-

gestion Sagredo you objected that everything outside an observerrsquos horizon would

naturally be part of his environment and would be gone from our description if we

trace over it

Sagredo which means that the whole global description would be gone

Salviati but why is that a problem No observer inside the universe can ever

see more than what is in their past light-cone at late times or more precisely in their

causal diamond We may not be able to recover the global picture by tracing over

the region behind an observerrsquos horizon but the same procedure might well achieve

decoherence in the region the observer can actually access In fact we donrsquot even

need an actual observer we can get decoherence by tracing over degrees of freedom

that leave the causal horizon of any worldline This will allow us to say that a bubble

formed in one place and not another So why donrsquot we give up on the global description

for a moment Later on we can check whether a global picture can be recovered in

some way from the decoherent causal diamonds

Salviati hands out Sections 24ndash26

24 Objective decoherence from the causal diamond

If Hawking radiation contains the full information about the quantum state of a star

that collapsed to form a black hole then there is an apparent paradox The star is

located inside the black hole at spacelike separation from the Hawking cloud hence two

copies of the original quantum information are present simultaneously The xeroxing of

quantum information however conflicts with the linearity of quantum mechanics [19]

ndash 12 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

periences of that observer But this subjective viewpoint clashes with the impersonal

unitary description of large spacetime regionsmdashthe viewpoint usually adopted in cos-

mology We are free of course to pick any subsystem and trace over it But the

outcome will depend on this choice The usual choices implicitly involve locality but

not in a unique way

For example we might choose S to be an electron and E to be the inanimate

laboratory The systemrsquos wave function collapses when the electron becomes entangled

with some detector But we may also include in S everything out to the edge of the

solar system The environment is whatever is out beyond the orbit of Pluto In that

case the collapse of the system wavefunction cannot take place until a photon from the

detector has passed Plutorsquos orbit This would take about a five hours during which the

system wavefunction is coherent

In particular decoherence cannot occur in the complete quantum description of

any region larger than the future light-cone of the measurement event M (Fig 1) All

environmental degrees of freedom that could have become entangled with the apparatus

since the measurement took place must lie within this lightcone and hence are included

not traced over in a complete description of the state An example of such a region

is the whole universe ie any Cauchy surface to the future of M But at least at

sufficiently early times the future light-cone of M will be much smaller than the whole

universe Already on this scale the system SAE will be coherent

In our earlier example suppose that we measure the spin of an electron that is

initially prepared in a superposition of spin-up and spin-down a|0〉S + b|1〉S resulting

in the state |ψ〉 of Eq (12) A complete description of the solar system (defined as

the interior of a sphere the size of Plutorsquos orbit with a light-crossing time of about

10 hours) by a local quantum field theory contains every particle that could possibly

have interacted with the apparatus after the measurement for about 5 hours This

description would maintain the coherence of the macroscopic superpositions implicit in

the state |ψ〉 such as apparatus-up with apparatus-down until the first photons that

are entangled with the apparatus leave the solar system

Of course a detailed knowledge of the quantum state in such large regions is

unavailable to a realistic observer (Indeed if the region is larger than a cosmological

event horizon then its quantum state is cannot be probed at all without violating

causality) Yet our theoretical description of matter fields in spacetime retains in

principle all degrees of freedom and full coherence of the quantum state In theoretical

cosmology this can lead to inconsistencies if we describe regions that are larger than the

future light-cones of events that we nevertheless treat as decohered We now consider

an important example

ndash 7 ndash

22 Failure to decohere A problem with the global multiverse

The above analysis undermines what we will call the ldquostandard global picturerdquo of an

eternally inflating spacetime Consider an effective potential containing at least one

inflating false vacuum ie a metastable de Sitter vacuum with decay rate much less

than one decay per Hubble volume and Hubble time We will also assume that there

is at least one terminal vacuum with nonpositive cosmological constant (The string

theory landscape is believed to have at least 10100primes of vacua of both types [14ndash17])

According to the standard description of eternal inflation an inflating vacuum nu-

cleates bubble-universes in a statistical manner similar to the way superheated water

nucleates bubbles of steam That process is described by classical stochastic production

of bubbles which occurs randomly but the randomness is classical The bubbles nucle-

ate at definite locations and coherent quantum mechanical interference plays no role

The conventional description of eternal inflation similarly based on classical stochastic

processes However this picture is not consistent with a complete quantum-mechanical

description of a global region of the multiverse

To explain why this is so consider the future domain of dependence D(Σ0) of a

sufficiently large hypersurface Σ0 which need not be a Cauchy surface D(Σ0) consists

of all events that can be predicted from data on Σ0 see Fig 2 If Σ0 contains suffi-

ciently large and long-lived metastable de Sitter regions then bubbles of vacua of lower

energy do not consume the parent de Sitter vacua in which they nucleate [18] Hence

the de Sitter vacua are said to inflate eternally producing an unbounded number of

bubble universes The resulting spacetime is said to have the structure shown in the

conformal diagram in Fig 2 with bubbles nucleating at definite spacetime events The

future conformal boundary is spacelike in regions with negative cosmological constant

corresponding to a local big crunch The boundary contains null ldquohatsrdquo in regions

occupied by vacua with Λ = 0

But this picture does not arise in a complete quantum description of D(Σ0) The

future light-cones of events at late times are much smaller than D(Σ0) In any state

that describes the entire spacetime region D(Σ0) decoherence can only take place at

the margin of D(Σ0) (shown light shaded in Fig 2) in the region from which particles

can escape into the complement of D(Σ0) in the full spacetime No decoherence can

take place in the infinite spacetime region defined by the past domain of dependence

of the future boundary of D(Σ0) In this region quantum evolution remains coherent

even if it results in the superposition of macroscopically distinct matter or spacetime

configurations

An important example is the superposition of vacuum decays taking place at dif-

ferent places Without decoherence it makes no sense to say that bubbles nucleate at

ndash 8 ndash

future boundary

Σ0

Figure 2 The future domain of dependence D(Σ0) (light or dark shaded) is the spacetime

region that can be predicted from data on the timeslice Σ0 If the future conformal boundary

contains spacelike portions as in eternal inflation or inside a black hole then the future

light-cones of events in the dark shaded region remain entirely within D(Σ0) Pure quantum

states do not decohere in this region in a complete description of D(Σ0) This is true even for

states that involve macroscopic superpositions such as the locations of pocket universes in

eternal inflation (dashed lines) calling into question the self-consistency of the global picture

of eternal inflation

particular times and locations rather a wavefunction with initial support only in the

parent vacuum develops into a superposition of parent and daughter vacua Bubbles

nucleating at all places at times are ldquoquantum superimposedrdquo With the gravitational

backreaction included the metric too would remain in a quantum-mechanical super-

position This contradicts the standard global picture of eternal inflation in which

domain walls vacua and the spacetime metric take on definite values as if drawn from

a density matrix obtained by tracing over some degrees of freedom and as if the inter-

action with these degrees of freedom had picked out a preferred basis that eliminates

the quantum superposition of bubbles and vacua

Let us quickly get rid of one red herring Can the standard geometry of eternal

inflation be recovered by using so-called semi-classical gravity in which the metric is

sourced by the expectation value of the energy-momentum tensor

Gmicroν = 8π〈Tmicroν〉 (21)

This does not work because the matter quantum fields would still remain coherent At

the level of the quantum fields the wavefunction initially has support only in the false

vacuum Over time it evolves to a superposition of the false vacuum (with decreasing

amplitude) with the true vacuum (with increasing amplitude) plus a superposition

of expanding and colliding domain walls This state is quite complicated but the

expectation value of its stress tensor should remain spatially homogeneous if it was so

initially The net effect over time would be a continuous conversion of vacuum energy

into ordinary matter or radiation (from the collision of bubbles and motion of the scalar

field) By Eq (21) the geometry spacetime would respond to the homogeneous glide

ndash 9 ndash

of the vacuum energy to negative values This would result in a global crunch after

finite time in stark contrast to the standard picture of global eternal inflation In

any case it seems implausible that semi-classical gravity should apply in a situation in

which coherent branches of the wavefunction have radically different gravitational back-

reaction The AdSCFT correspondence provides an explicit counterexample since the

superposition of two CFT states that correspond to different classical geometries must

correspond to a quantum superposition of the two metrics

The conclusion that we come to from these considerations is not that the global

multiverse is meaningless but that the parallel view should not be implemented by

unitary quantum mechanics But is there an alternative Can the standard global

picture be recovered by considering an observer who has access only to some of the

degrees of freedom of the multiverse and appealing to decoherence We debate this

question in the following section

23 Simpliciorsquos proposal

Simplicio and Sagredo have studied Sections 21 and 22 supplied to them by Salviati

They meet at Sagredorsquos house for a discussion

Simplicio You have convinced me that a complete description of eternal inflation

by unitary quantum evolution on global slices will not lead to a picture in which bubbles

form at definite places and times But all I need is an observer somewhere Then I

can take this observerrsquos point of view and trace over the degrees of freedom that are

inaccessible to him This decoheres events such as bubble nucleations in the entire

global multiverse It actually helps that some regions are causally disconnected from the

observer this makes his environmentmdashthe degrees of freedom he fails to accessmdashreally

huge

Sagredo An interesting idea But you seem to include everything outside the

observerrsquos horizon region in what you call the environment Once you trace over it it

is gone from your description and you could not possibly recover a global spacetime

Simplicio Your objection is valid but it also shows me how to fix my proposal

The observer should only trace over environmental degrees in his own horizon Deco-

herence is very efficient so this should suffice

Sagredo I wonder what would happen if there were two observers in widely

separated regions If one observerrsquos environment is enough to decohere the whole

universe which one should we pick

Simplicio I have not done a calculation but it seems to me that it shouldnrsquot

matter The outcome of an experiment by one of the observers should be the same no

matter which observerrsquos environment I trace over That is certainly how it works when

you and I both stare at the same apparatus

ndash 10 ndash

future boundary

Σ0

P O

Figure 3 Environmental degrees of freedom entangled with an observer at O remain within

the causal future of the causal past of O J+[Jminus(O)] (cyanshaded) They are not entangled

with distant regions of the multiverse Tracing over them will not lead to decoherence of

a bubble nucleated at P for example and hence will fail to reproduce the standard global

picture of eternal inflation

Sagredo Something is different about the multiverse When you and I both

observe Salviati we all become correlated by interactions with a common environment

But how does an observer in one horizon volume become correlated with an object in

another horizon volume far away

Salviati Sagredo you hit the nail on the head Decoherence requires the inter-

action of environmental degrees of freedom with the apparatus and the observer This

entangles them and it leads to a density matrix once the environment is ignored by the

observer But an observer cannot have interacted with degrees of freedom that were

never present in his past light-cone

Sagredo Thank you for articulating so clearly what to me was only a vague

concern Simplicio you look puzzled so let me summarize our objection in my own

words You proposed a method for obtaining the standard global picture of eternal

inflation you claim that we need only identify an arbitrary observer in the multiverse

and trace over his environment If we defined the environment as all degrees of freedom

the observer fails to monitor then it would include the causally disconnected regions

outside his horizon With this definition these regions will disappear entirely from your

description in conflict with the global picture So we agreed to define the environment

as the degrees of freedom that have interacted with the observer and which he cannot

access in practice But in this case the environment includes no degrees of freedom

outside the causal future of the observerrsquos causal past I have drawn this region in

Fig 3 But tracing over an environment can only decohere degrees of freedom that it

is entangled with In this case it can decohere some events that lie in the observerrsquos

past light-cone But it cannot affect quantum coherence in far-away horizon regions

because the environment you have picked is not entangled with these regions In those

ndash 11 ndash

regions bubble walls and vacua will remain in superposition which again conflicts with

the standard global picture of eternal inflation

Simplicio I see that my idea still has some problems I will need to identify more

than one observer-environment pair In fact if I wish to preserve the global picture

of the multiverse I will have to assume that an observer is present in every horizon

volume at all times Otherwise there will be horizon regions where no one is around

to decide which degrees of freedom are hard to keep track of so there is no way to

identify and trace over an environment In such regions bubbles would not form at

particular places and times in conflict with the standard global picture

Sagredo But this assumption is clearly violated in many landscape models Most

de Sitter vacua have large cosmological constant so that a single horizon volume is too

small to contain the large number of degrees of freedom required for an observer And

regions with small vacuum energy may be very long lived so the corresponding bubbles

contain many horizon volumes that are completely empty Irsquom afraid Simplicio that

your efforts to rescue the global multiverse are destined to fail

Salviati Why donrsquot we back up a little and return to Simpliciorsquos initial sug-

gestion Sagredo you objected that everything outside an observerrsquos horizon would

naturally be part of his environment and would be gone from our description if we

trace over it

Sagredo which means that the whole global description would be gone

Salviati but why is that a problem No observer inside the universe can ever

see more than what is in their past light-cone at late times or more precisely in their

causal diamond We may not be able to recover the global picture by tracing over

the region behind an observerrsquos horizon but the same procedure might well achieve

decoherence in the region the observer can actually access In fact we donrsquot even

need an actual observer we can get decoherence by tracing over degrees of freedom

that leave the causal horizon of any worldline This will allow us to say that a bubble

formed in one place and not another So why donrsquot we give up on the global description

for a moment Later on we can check whether a global picture can be recovered in

some way from the decoherent causal diamonds

Salviati hands out Sections 24ndash26

24 Objective decoherence from the causal diamond

If Hawking radiation contains the full information about the quantum state of a star

that collapsed to form a black hole then there is an apparent paradox The star is

located inside the black hole at spacelike separation from the Hawking cloud hence two

copies of the original quantum information are present simultaneously The xeroxing of

quantum information however conflicts with the linearity of quantum mechanics [19]

ndash 12 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

22 Failure to decohere A problem with the global multiverse

The above analysis undermines what we will call the ldquostandard global picturerdquo of an

eternally inflating spacetime Consider an effective potential containing at least one

inflating false vacuum ie a metastable de Sitter vacuum with decay rate much less

than one decay per Hubble volume and Hubble time We will also assume that there

is at least one terminal vacuum with nonpositive cosmological constant (The string

theory landscape is believed to have at least 10100primes of vacua of both types [14ndash17])

According to the standard description of eternal inflation an inflating vacuum nu-

cleates bubble-universes in a statistical manner similar to the way superheated water

nucleates bubbles of steam That process is described by classical stochastic production

of bubbles which occurs randomly but the randomness is classical The bubbles nucle-

ate at definite locations and coherent quantum mechanical interference plays no role

The conventional description of eternal inflation similarly based on classical stochastic

processes However this picture is not consistent with a complete quantum-mechanical

description of a global region of the multiverse

To explain why this is so consider the future domain of dependence D(Σ0) of a

sufficiently large hypersurface Σ0 which need not be a Cauchy surface D(Σ0) consists

of all events that can be predicted from data on Σ0 see Fig 2 If Σ0 contains suffi-

ciently large and long-lived metastable de Sitter regions then bubbles of vacua of lower

energy do not consume the parent de Sitter vacua in which they nucleate [18] Hence

the de Sitter vacua are said to inflate eternally producing an unbounded number of

bubble universes The resulting spacetime is said to have the structure shown in the

conformal diagram in Fig 2 with bubbles nucleating at definite spacetime events The

future conformal boundary is spacelike in regions with negative cosmological constant

corresponding to a local big crunch The boundary contains null ldquohatsrdquo in regions

occupied by vacua with Λ = 0

But this picture does not arise in a complete quantum description of D(Σ0) The

future light-cones of events at late times are much smaller than D(Σ0) In any state

that describes the entire spacetime region D(Σ0) decoherence can only take place at

the margin of D(Σ0) (shown light shaded in Fig 2) in the region from which particles

can escape into the complement of D(Σ0) in the full spacetime No decoherence can

take place in the infinite spacetime region defined by the past domain of dependence

of the future boundary of D(Σ0) In this region quantum evolution remains coherent

even if it results in the superposition of macroscopically distinct matter or spacetime

configurations

An important example is the superposition of vacuum decays taking place at dif-

ferent places Without decoherence it makes no sense to say that bubbles nucleate at

ndash 8 ndash

future boundary

Σ0

Figure 2 The future domain of dependence D(Σ0) (light or dark shaded) is the spacetime

region that can be predicted from data on the timeslice Σ0 If the future conformal boundary

contains spacelike portions as in eternal inflation or inside a black hole then the future

light-cones of events in the dark shaded region remain entirely within D(Σ0) Pure quantum

states do not decohere in this region in a complete description of D(Σ0) This is true even for

states that involve macroscopic superpositions such as the locations of pocket universes in

eternal inflation (dashed lines) calling into question the self-consistency of the global picture

of eternal inflation

particular times and locations rather a wavefunction with initial support only in the

parent vacuum develops into a superposition of parent and daughter vacua Bubbles

nucleating at all places at times are ldquoquantum superimposedrdquo With the gravitational

backreaction included the metric too would remain in a quantum-mechanical super-

position This contradicts the standard global picture of eternal inflation in which

domain walls vacua and the spacetime metric take on definite values as if drawn from

a density matrix obtained by tracing over some degrees of freedom and as if the inter-

action with these degrees of freedom had picked out a preferred basis that eliminates

the quantum superposition of bubbles and vacua

Let us quickly get rid of one red herring Can the standard geometry of eternal

inflation be recovered by using so-called semi-classical gravity in which the metric is

sourced by the expectation value of the energy-momentum tensor

Gmicroν = 8π〈Tmicroν〉 (21)

This does not work because the matter quantum fields would still remain coherent At

the level of the quantum fields the wavefunction initially has support only in the false

vacuum Over time it evolves to a superposition of the false vacuum (with decreasing

amplitude) with the true vacuum (with increasing amplitude) plus a superposition

of expanding and colliding domain walls This state is quite complicated but the

expectation value of its stress tensor should remain spatially homogeneous if it was so

initially The net effect over time would be a continuous conversion of vacuum energy

into ordinary matter or radiation (from the collision of bubbles and motion of the scalar

field) By Eq (21) the geometry spacetime would respond to the homogeneous glide

ndash 9 ndash

of the vacuum energy to negative values This would result in a global crunch after

finite time in stark contrast to the standard picture of global eternal inflation In

any case it seems implausible that semi-classical gravity should apply in a situation in

which coherent branches of the wavefunction have radically different gravitational back-

reaction The AdSCFT correspondence provides an explicit counterexample since the

superposition of two CFT states that correspond to different classical geometries must

correspond to a quantum superposition of the two metrics

The conclusion that we come to from these considerations is not that the global

multiverse is meaningless but that the parallel view should not be implemented by

unitary quantum mechanics But is there an alternative Can the standard global

picture be recovered by considering an observer who has access only to some of the

degrees of freedom of the multiverse and appealing to decoherence We debate this

question in the following section

23 Simpliciorsquos proposal

Simplicio and Sagredo have studied Sections 21 and 22 supplied to them by Salviati

They meet at Sagredorsquos house for a discussion

Simplicio You have convinced me that a complete description of eternal inflation

by unitary quantum evolution on global slices will not lead to a picture in which bubbles

form at definite places and times But all I need is an observer somewhere Then I

can take this observerrsquos point of view and trace over the degrees of freedom that are

inaccessible to him This decoheres events such as bubble nucleations in the entire

global multiverse It actually helps that some regions are causally disconnected from the

observer this makes his environmentmdashthe degrees of freedom he fails to accessmdashreally

huge

Sagredo An interesting idea But you seem to include everything outside the

observerrsquos horizon region in what you call the environment Once you trace over it it

is gone from your description and you could not possibly recover a global spacetime

Simplicio Your objection is valid but it also shows me how to fix my proposal

The observer should only trace over environmental degrees in his own horizon Deco-

herence is very efficient so this should suffice

Sagredo I wonder what would happen if there were two observers in widely

separated regions If one observerrsquos environment is enough to decohere the whole

universe which one should we pick

Simplicio I have not done a calculation but it seems to me that it shouldnrsquot

matter The outcome of an experiment by one of the observers should be the same no

matter which observerrsquos environment I trace over That is certainly how it works when

you and I both stare at the same apparatus

ndash 10 ndash

future boundary

Σ0

P O

Figure 3 Environmental degrees of freedom entangled with an observer at O remain within

the causal future of the causal past of O J+[Jminus(O)] (cyanshaded) They are not entangled

with distant regions of the multiverse Tracing over them will not lead to decoherence of

a bubble nucleated at P for example and hence will fail to reproduce the standard global

picture of eternal inflation

Sagredo Something is different about the multiverse When you and I both

observe Salviati we all become correlated by interactions with a common environment

But how does an observer in one horizon volume become correlated with an object in

another horizon volume far away

Salviati Sagredo you hit the nail on the head Decoherence requires the inter-

action of environmental degrees of freedom with the apparatus and the observer This

entangles them and it leads to a density matrix once the environment is ignored by the

observer But an observer cannot have interacted with degrees of freedom that were

never present in his past light-cone

Sagredo Thank you for articulating so clearly what to me was only a vague

concern Simplicio you look puzzled so let me summarize our objection in my own

words You proposed a method for obtaining the standard global picture of eternal

inflation you claim that we need only identify an arbitrary observer in the multiverse

and trace over his environment If we defined the environment as all degrees of freedom

the observer fails to monitor then it would include the causally disconnected regions

outside his horizon With this definition these regions will disappear entirely from your

description in conflict with the global picture So we agreed to define the environment

as the degrees of freedom that have interacted with the observer and which he cannot

access in practice But in this case the environment includes no degrees of freedom

outside the causal future of the observerrsquos causal past I have drawn this region in

Fig 3 But tracing over an environment can only decohere degrees of freedom that it

is entangled with In this case it can decohere some events that lie in the observerrsquos

past light-cone But it cannot affect quantum coherence in far-away horizon regions

because the environment you have picked is not entangled with these regions In those

ndash 11 ndash

regions bubble walls and vacua will remain in superposition which again conflicts with

the standard global picture of eternal inflation

Simplicio I see that my idea still has some problems I will need to identify more

than one observer-environment pair In fact if I wish to preserve the global picture

of the multiverse I will have to assume that an observer is present in every horizon

volume at all times Otherwise there will be horizon regions where no one is around

to decide which degrees of freedom are hard to keep track of so there is no way to

identify and trace over an environment In such regions bubbles would not form at

particular places and times in conflict with the standard global picture

Sagredo But this assumption is clearly violated in many landscape models Most

de Sitter vacua have large cosmological constant so that a single horizon volume is too

small to contain the large number of degrees of freedom required for an observer And

regions with small vacuum energy may be very long lived so the corresponding bubbles

contain many horizon volumes that are completely empty Irsquom afraid Simplicio that

your efforts to rescue the global multiverse are destined to fail

Salviati Why donrsquot we back up a little and return to Simpliciorsquos initial sug-

gestion Sagredo you objected that everything outside an observerrsquos horizon would

naturally be part of his environment and would be gone from our description if we

trace over it

Sagredo which means that the whole global description would be gone

Salviati but why is that a problem No observer inside the universe can ever

see more than what is in their past light-cone at late times or more precisely in their

causal diamond We may not be able to recover the global picture by tracing over

the region behind an observerrsquos horizon but the same procedure might well achieve

decoherence in the region the observer can actually access In fact we donrsquot even

need an actual observer we can get decoherence by tracing over degrees of freedom

that leave the causal horizon of any worldline This will allow us to say that a bubble

formed in one place and not another So why donrsquot we give up on the global description

for a moment Later on we can check whether a global picture can be recovered in

some way from the decoherent causal diamonds

Salviati hands out Sections 24ndash26

24 Objective decoherence from the causal diamond

If Hawking radiation contains the full information about the quantum state of a star

that collapsed to form a black hole then there is an apparent paradox The star is

located inside the black hole at spacelike separation from the Hawking cloud hence two

copies of the original quantum information are present simultaneously The xeroxing of

quantum information however conflicts with the linearity of quantum mechanics [19]

ndash 12 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

future boundary

Σ0

Figure 2 The future domain of dependence D(Σ0) (light or dark shaded) is the spacetime

region that can be predicted from data on the timeslice Σ0 If the future conformal boundary

contains spacelike portions as in eternal inflation or inside a black hole then the future

light-cones of events in the dark shaded region remain entirely within D(Σ0) Pure quantum

states do not decohere in this region in a complete description of D(Σ0) This is true even for

states that involve macroscopic superpositions such as the locations of pocket universes in

eternal inflation (dashed lines) calling into question the self-consistency of the global picture

of eternal inflation

particular times and locations rather a wavefunction with initial support only in the

parent vacuum develops into a superposition of parent and daughter vacua Bubbles

nucleating at all places at times are ldquoquantum superimposedrdquo With the gravitational

backreaction included the metric too would remain in a quantum-mechanical super-

position This contradicts the standard global picture of eternal inflation in which

domain walls vacua and the spacetime metric take on definite values as if drawn from

a density matrix obtained by tracing over some degrees of freedom and as if the inter-

action with these degrees of freedom had picked out a preferred basis that eliminates

the quantum superposition of bubbles and vacua

Let us quickly get rid of one red herring Can the standard geometry of eternal

inflation be recovered by using so-called semi-classical gravity in which the metric is

sourced by the expectation value of the energy-momentum tensor

Gmicroν = 8π〈Tmicroν〉 (21)

This does not work because the matter quantum fields would still remain coherent At

the level of the quantum fields the wavefunction initially has support only in the false

vacuum Over time it evolves to a superposition of the false vacuum (with decreasing

amplitude) with the true vacuum (with increasing amplitude) plus a superposition

of expanding and colliding domain walls This state is quite complicated but the

expectation value of its stress tensor should remain spatially homogeneous if it was so

initially The net effect over time would be a continuous conversion of vacuum energy

into ordinary matter or radiation (from the collision of bubbles and motion of the scalar

field) By Eq (21) the geometry spacetime would respond to the homogeneous glide

ndash 9 ndash

of the vacuum energy to negative values This would result in a global crunch after

finite time in stark contrast to the standard picture of global eternal inflation In

any case it seems implausible that semi-classical gravity should apply in a situation in

which coherent branches of the wavefunction have radically different gravitational back-

reaction The AdSCFT correspondence provides an explicit counterexample since the

superposition of two CFT states that correspond to different classical geometries must

correspond to a quantum superposition of the two metrics

The conclusion that we come to from these considerations is not that the global

multiverse is meaningless but that the parallel view should not be implemented by

unitary quantum mechanics But is there an alternative Can the standard global

picture be recovered by considering an observer who has access only to some of the

degrees of freedom of the multiverse and appealing to decoherence We debate this

question in the following section

23 Simpliciorsquos proposal

Simplicio and Sagredo have studied Sections 21 and 22 supplied to them by Salviati

They meet at Sagredorsquos house for a discussion

Simplicio You have convinced me that a complete description of eternal inflation

by unitary quantum evolution on global slices will not lead to a picture in which bubbles

form at definite places and times But all I need is an observer somewhere Then I

can take this observerrsquos point of view and trace over the degrees of freedom that are

inaccessible to him This decoheres events such as bubble nucleations in the entire

global multiverse It actually helps that some regions are causally disconnected from the

observer this makes his environmentmdashthe degrees of freedom he fails to accessmdashreally

huge

Sagredo An interesting idea But you seem to include everything outside the

observerrsquos horizon region in what you call the environment Once you trace over it it

is gone from your description and you could not possibly recover a global spacetime

Simplicio Your objection is valid but it also shows me how to fix my proposal

The observer should only trace over environmental degrees in his own horizon Deco-

herence is very efficient so this should suffice

Sagredo I wonder what would happen if there were two observers in widely

separated regions If one observerrsquos environment is enough to decohere the whole

universe which one should we pick

Simplicio I have not done a calculation but it seems to me that it shouldnrsquot

matter The outcome of an experiment by one of the observers should be the same no

matter which observerrsquos environment I trace over That is certainly how it works when

you and I both stare at the same apparatus

ndash 10 ndash

future boundary

Σ0

P O

Figure 3 Environmental degrees of freedom entangled with an observer at O remain within

the causal future of the causal past of O J+[Jminus(O)] (cyanshaded) They are not entangled

with distant regions of the multiverse Tracing over them will not lead to decoherence of

a bubble nucleated at P for example and hence will fail to reproduce the standard global

picture of eternal inflation

Sagredo Something is different about the multiverse When you and I both

observe Salviati we all become correlated by interactions with a common environment

But how does an observer in one horizon volume become correlated with an object in

another horizon volume far away

Salviati Sagredo you hit the nail on the head Decoherence requires the inter-

action of environmental degrees of freedom with the apparatus and the observer This

entangles them and it leads to a density matrix once the environment is ignored by the

observer But an observer cannot have interacted with degrees of freedom that were

never present in his past light-cone

Sagredo Thank you for articulating so clearly what to me was only a vague

concern Simplicio you look puzzled so let me summarize our objection in my own

words You proposed a method for obtaining the standard global picture of eternal

inflation you claim that we need only identify an arbitrary observer in the multiverse

and trace over his environment If we defined the environment as all degrees of freedom

the observer fails to monitor then it would include the causally disconnected regions

outside his horizon With this definition these regions will disappear entirely from your

description in conflict with the global picture So we agreed to define the environment

as the degrees of freedom that have interacted with the observer and which he cannot

access in practice But in this case the environment includes no degrees of freedom

outside the causal future of the observerrsquos causal past I have drawn this region in

Fig 3 But tracing over an environment can only decohere degrees of freedom that it

is entangled with In this case it can decohere some events that lie in the observerrsquos

past light-cone But it cannot affect quantum coherence in far-away horizon regions

because the environment you have picked is not entangled with these regions In those

ndash 11 ndash

regions bubble walls and vacua will remain in superposition which again conflicts with

the standard global picture of eternal inflation

Simplicio I see that my idea still has some problems I will need to identify more

than one observer-environment pair In fact if I wish to preserve the global picture

of the multiverse I will have to assume that an observer is present in every horizon

volume at all times Otherwise there will be horizon regions where no one is around

to decide which degrees of freedom are hard to keep track of so there is no way to

identify and trace over an environment In such regions bubbles would not form at

particular places and times in conflict with the standard global picture

Sagredo But this assumption is clearly violated in many landscape models Most

de Sitter vacua have large cosmological constant so that a single horizon volume is too

small to contain the large number of degrees of freedom required for an observer And

regions with small vacuum energy may be very long lived so the corresponding bubbles

contain many horizon volumes that are completely empty Irsquom afraid Simplicio that

your efforts to rescue the global multiverse are destined to fail

Salviati Why donrsquot we back up a little and return to Simpliciorsquos initial sug-

gestion Sagredo you objected that everything outside an observerrsquos horizon would

naturally be part of his environment and would be gone from our description if we

trace over it

Sagredo which means that the whole global description would be gone

Salviati but why is that a problem No observer inside the universe can ever

see more than what is in their past light-cone at late times or more precisely in their

causal diamond We may not be able to recover the global picture by tracing over

the region behind an observerrsquos horizon but the same procedure might well achieve

decoherence in the region the observer can actually access In fact we donrsquot even

need an actual observer we can get decoherence by tracing over degrees of freedom

that leave the causal horizon of any worldline This will allow us to say that a bubble

formed in one place and not another So why donrsquot we give up on the global description

for a moment Later on we can check whether a global picture can be recovered in

some way from the decoherent causal diamonds

Salviati hands out Sections 24ndash26

24 Objective decoherence from the causal diamond

If Hawking radiation contains the full information about the quantum state of a star

that collapsed to form a black hole then there is an apparent paradox The star is

located inside the black hole at spacelike separation from the Hawking cloud hence two

copies of the original quantum information are present simultaneously The xeroxing of

quantum information however conflicts with the linearity of quantum mechanics [19]

ndash 12 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

of the vacuum energy to negative values This would result in a global crunch after

finite time in stark contrast to the standard picture of global eternal inflation In

any case it seems implausible that semi-classical gravity should apply in a situation in

which coherent branches of the wavefunction have radically different gravitational back-

reaction The AdSCFT correspondence provides an explicit counterexample since the

superposition of two CFT states that correspond to different classical geometries must

correspond to a quantum superposition of the two metrics

The conclusion that we come to from these considerations is not that the global

multiverse is meaningless but that the parallel view should not be implemented by

unitary quantum mechanics But is there an alternative Can the standard global

picture be recovered by considering an observer who has access only to some of the

degrees of freedom of the multiverse and appealing to decoherence We debate this

question in the following section

23 Simpliciorsquos proposal

Simplicio and Sagredo have studied Sections 21 and 22 supplied to them by Salviati

They meet at Sagredorsquos house for a discussion

Simplicio You have convinced me that a complete description of eternal inflation

by unitary quantum evolution on global slices will not lead to a picture in which bubbles

form at definite places and times But all I need is an observer somewhere Then I

can take this observerrsquos point of view and trace over the degrees of freedom that are

inaccessible to him This decoheres events such as bubble nucleations in the entire

global multiverse It actually helps that some regions are causally disconnected from the

observer this makes his environmentmdashthe degrees of freedom he fails to accessmdashreally

huge

Sagredo An interesting idea But you seem to include everything outside the

observerrsquos horizon region in what you call the environment Once you trace over it it

is gone from your description and you could not possibly recover a global spacetime

Simplicio Your objection is valid but it also shows me how to fix my proposal

The observer should only trace over environmental degrees in his own horizon Deco-

herence is very efficient so this should suffice

Sagredo I wonder what would happen if there were two observers in widely

separated regions If one observerrsquos environment is enough to decohere the whole

universe which one should we pick

Simplicio I have not done a calculation but it seems to me that it shouldnrsquot

matter The outcome of an experiment by one of the observers should be the same no

matter which observerrsquos environment I trace over That is certainly how it works when

you and I both stare at the same apparatus

ndash 10 ndash

future boundary

Σ0

P O

Figure 3 Environmental degrees of freedom entangled with an observer at O remain within

the causal future of the causal past of O J+[Jminus(O)] (cyanshaded) They are not entangled

with distant regions of the multiverse Tracing over them will not lead to decoherence of

a bubble nucleated at P for example and hence will fail to reproduce the standard global

picture of eternal inflation

Sagredo Something is different about the multiverse When you and I both

observe Salviati we all become correlated by interactions with a common environment

But how does an observer in one horizon volume become correlated with an object in

another horizon volume far away

Salviati Sagredo you hit the nail on the head Decoherence requires the inter-

action of environmental degrees of freedom with the apparatus and the observer This

entangles them and it leads to a density matrix once the environment is ignored by the

observer But an observer cannot have interacted with degrees of freedom that were

never present in his past light-cone

Sagredo Thank you for articulating so clearly what to me was only a vague

concern Simplicio you look puzzled so let me summarize our objection in my own

words You proposed a method for obtaining the standard global picture of eternal

inflation you claim that we need only identify an arbitrary observer in the multiverse

and trace over his environment If we defined the environment as all degrees of freedom

the observer fails to monitor then it would include the causally disconnected regions

outside his horizon With this definition these regions will disappear entirely from your

description in conflict with the global picture So we agreed to define the environment

as the degrees of freedom that have interacted with the observer and which he cannot

access in practice But in this case the environment includes no degrees of freedom

outside the causal future of the observerrsquos causal past I have drawn this region in

Fig 3 But tracing over an environment can only decohere degrees of freedom that it

is entangled with In this case it can decohere some events that lie in the observerrsquos

past light-cone But it cannot affect quantum coherence in far-away horizon regions

because the environment you have picked is not entangled with these regions In those

ndash 11 ndash

regions bubble walls and vacua will remain in superposition which again conflicts with

the standard global picture of eternal inflation

Simplicio I see that my idea still has some problems I will need to identify more

than one observer-environment pair In fact if I wish to preserve the global picture

of the multiverse I will have to assume that an observer is present in every horizon

volume at all times Otherwise there will be horizon regions where no one is around

to decide which degrees of freedom are hard to keep track of so there is no way to

identify and trace over an environment In such regions bubbles would not form at

particular places and times in conflict with the standard global picture

Sagredo But this assumption is clearly violated in many landscape models Most

de Sitter vacua have large cosmological constant so that a single horizon volume is too

small to contain the large number of degrees of freedom required for an observer And

regions with small vacuum energy may be very long lived so the corresponding bubbles

contain many horizon volumes that are completely empty Irsquom afraid Simplicio that

your efforts to rescue the global multiverse are destined to fail

Salviati Why donrsquot we back up a little and return to Simpliciorsquos initial sug-

gestion Sagredo you objected that everything outside an observerrsquos horizon would

naturally be part of his environment and would be gone from our description if we

trace over it

Sagredo which means that the whole global description would be gone

Salviati but why is that a problem No observer inside the universe can ever

see more than what is in their past light-cone at late times or more precisely in their

causal diamond We may not be able to recover the global picture by tracing over

the region behind an observerrsquos horizon but the same procedure might well achieve

decoherence in the region the observer can actually access In fact we donrsquot even

need an actual observer we can get decoherence by tracing over degrees of freedom

that leave the causal horizon of any worldline This will allow us to say that a bubble

formed in one place and not another So why donrsquot we give up on the global description

for a moment Later on we can check whether a global picture can be recovered in

some way from the decoherent causal diamonds

Salviati hands out Sections 24ndash26

24 Objective decoherence from the causal diamond

If Hawking radiation contains the full information about the quantum state of a star

that collapsed to form a black hole then there is an apparent paradox The star is

located inside the black hole at spacelike separation from the Hawking cloud hence two

copies of the original quantum information are present simultaneously The xeroxing of

quantum information however conflicts with the linearity of quantum mechanics [19]

ndash 12 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

future boundary

Σ0

P O

Figure 3 Environmental degrees of freedom entangled with an observer at O remain within

the causal future of the causal past of O J+[Jminus(O)] (cyanshaded) They are not entangled

with distant regions of the multiverse Tracing over them will not lead to decoherence of

a bubble nucleated at P for example and hence will fail to reproduce the standard global

picture of eternal inflation

Sagredo Something is different about the multiverse When you and I both

observe Salviati we all become correlated by interactions with a common environment

But how does an observer in one horizon volume become correlated with an object in

another horizon volume far away

Salviati Sagredo you hit the nail on the head Decoherence requires the inter-

action of environmental degrees of freedom with the apparatus and the observer This

entangles them and it leads to a density matrix once the environment is ignored by the

observer But an observer cannot have interacted with degrees of freedom that were

never present in his past light-cone

Sagredo Thank you for articulating so clearly what to me was only a vague

concern Simplicio you look puzzled so let me summarize our objection in my own

words You proposed a method for obtaining the standard global picture of eternal

inflation you claim that we need only identify an arbitrary observer in the multiverse

and trace over his environment If we defined the environment as all degrees of freedom

the observer fails to monitor then it would include the causally disconnected regions

outside his horizon With this definition these regions will disappear entirely from your

description in conflict with the global picture So we agreed to define the environment

as the degrees of freedom that have interacted with the observer and which he cannot

access in practice But in this case the environment includes no degrees of freedom

outside the causal future of the observerrsquos causal past I have drawn this region in

Fig 3 But tracing over an environment can only decohere degrees of freedom that it

is entangled with In this case it can decohere some events that lie in the observerrsquos

past light-cone But it cannot affect quantum coherence in far-away horizon regions

because the environment you have picked is not entangled with these regions In those

ndash 11 ndash

regions bubble walls and vacua will remain in superposition which again conflicts with

the standard global picture of eternal inflation

Simplicio I see that my idea still has some problems I will need to identify more

than one observer-environment pair In fact if I wish to preserve the global picture

of the multiverse I will have to assume that an observer is present in every horizon

volume at all times Otherwise there will be horizon regions where no one is around

to decide which degrees of freedom are hard to keep track of so there is no way to

identify and trace over an environment In such regions bubbles would not form at

particular places and times in conflict with the standard global picture

Sagredo But this assumption is clearly violated in many landscape models Most

de Sitter vacua have large cosmological constant so that a single horizon volume is too

small to contain the large number of degrees of freedom required for an observer And

regions with small vacuum energy may be very long lived so the corresponding bubbles

contain many horizon volumes that are completely empty Irsquom afraid Simplicio that

your efforts to rescue the global multiverse are destined to fail

Salviati Why donrsquot we back up a little and return to Simpliciorsquos initial sug-

gestion Sagredo you objected that everything outside an observerrsquos horizon would

naturally be part of his environment and would be gone from our description if we

trace over it

Sagredo which means that the whole global description would be gone

Salviati but why is that a problem No observer inside the universe can ever

see more than what is in their past light-cone at late times or more precisely in their

causal diamond We may not be able to recover the global picture by tracing over

the region behind an observerrsquos horizon but the same procedure might well achieve

decoherence in the region the observer can actually access In fact we donrsquot even

need an actual observer we can get decoherence by tracing over degrees of freedom

that leave the causal horizon of any worldline This will allow us to say that a bubble

formed in one place and not another So why donrsquot we give up on the global description

for a moment Later on we can check whether a global picture can be recovered in

some way from the decoherent causal diamonds

Salviati hands out Sections 24ndash26

24 Objective decoherence from the causal diamond

If Hawking radiation contains the full information about the quantum state of a star

that collapsed to form a black hole then there is an apparent paradox The star is

located inside the black hole at spacelike separation from the Hawking cloud hence two

copies of the original quantum information are present simultaneously The xeroxing of

quantum information however conflicts with the linearity of quantum mechanics [19]

ndash 12 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

regions bubble walls and vacua will remain in superposition which again conflicts with

the standard global picture of eternal inflation

Simplicio I see that my idea still has some problems I will need to identify more

than one observer-environment pair In fact if I wish to preserve the global picture

of the multiverse I will have to assume that an observer is present in every horizon

volume at all times Otherwise there will be horizon regions where no one is around

to decide which degrees of freedom are hard to keep track of so there is no way to

identify and trace over an environment In such regions bubbles would not form at

particular places and times in conflict with the standard global picture

Sagredo But this assumption is clearly violated in many landscape models Most

de Sitter vacua have large cosmological constant so that a single horizon volume is too

small to contain the large number of degrees of freedom required for an observer And

regions with small vacuum energy may be very long lived so the corresponding bubbles

contain many horizon volumes that are completely empty Irsquom afraid Simplicio that

your efforts to rescue the global multiverse are destined to fail

Salviati Why donrsquot we back up a little and return to Simpliciorsquos initial sug-

gestion Sagredo you objected that everything outside an observerrsquos horizon would

naturally be part of his environment and would be gone from our description if we

trace over it

Sagredo which means that the whole global description would be gone

Salviati but why is that a problem No observer inside the universe can ever

see more than what is in their past light-cone at late times or more precisely in their

causal diamond We may not be able to recover the global picture by tracing over

the region behind an observerrsquos horizon but the same procedure might well achieve

decoherence in the region the observer can actually access In fact we donrsquot even

need an actual observer we can get decoherence by tracing over degrees of freedom

that leave the causal horizon of any worldline This will allow us to say that a bubble

formed in one place and not another So why donrsquot we give up on the global description

for a moment Later on we can check whether a global picture can be recovered in

some way from the decoherent causal diamonds

Salviati hands out Sections 24ndash26

24 Objective decoherence from the causal diamond

If Hawking radiation contains the full information about the quantum state of a star

that collapsed to form a black hole then there is an apparent paradox The star is

located inside the black hole at spacelike separation from the Hawking cloud hence two

copies of the original quantum information are present simultaneously The xeroxing of

quantum information however conflicts with the linearity of quantum mechanics [19]

ndash 12 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

future boundary

Σ0

q

p

Figure 4 The causal diamond (pinkshaded) spanned by two events p and q is the set of

points that lie on causal curves from p to q p is called the origin and q the tip of the causal

diamond In the example shown p lies on the initial surface and q on the future conformal

boundary of the spacetime The causal diamond is largest spacetime region that can be

causally probed by an observer travelling from p to q

The paradox is resolved by ldquoblack hole complementarityrdquo [20] By causality no observer

can see both copies of the information A theory of everything should be able to describe

any experiment that can actually be performed by some observer in the universe but it

need not describe the global viewpoint of a ldquosuperobserverrdquo who sees both the interior

and the exterior of a black hole Evidently the global description is inconsistent and

must be rejected

If the global viewpoint fails in a black hole geometry then it must be abandoned

in any spacetime Hence it is useful to characterize generally what spacetime regions

can be causally probed An experiment beginning at a spacetime event p and ending at

the event q in the future of p can probe the causal diamond I+(p)cap Iminus(q) (Fig 4) By

starting earlier or finishing later the causal diamond can be enlarged In spacetimes

with a spacelike future boundary such as black holes and many cosmological solutions

the global universe is much larger than any causal diamond it contains Here we will be

interested in diamonds that are as large as possible in the sense that p and q correspond

to the past and future endpoints of an inextendible worldline

We will now argue that the causal diamond can play a useful role in making decoher-

ence more objective Our discussion will be completely general though for concreteness

it can be useful to think of causal diamonds in a landscape which start in a de Sitter

vacuum and end up after a number of decays in a crunching Λ lt 0 vacuum

Consider a causal diamond C with future boundary B and past boundary B as

shown in Fig 5 For simplicity suppose that the initial state on B is pure Matter

degrees of freedom that leave the diamond by crossing B become inaccessible to any

ndash 13 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

Figure 5 Causal diamond spanned by the world-line (green) of an observer Environmental

degrees of freedom (purple dashed line) that leave the observerrsquos past light-cone (blue) at

some finite time can be recovered using mirrors

experiment within C by causality Therefore they must be traced over

In practice there will be many other degrees of freedom that an observer fails

to control including most degrees of freedom that have exited his past light-cone at

any finite time along his worldline But such degrees of freedom can be reflected by

mirrors or in some other way change their direction of motion back towards the observer

(Fig 5) Thus at least in principle the observer could later be brought into contact

again with any degrees of freedom that remain within the causal diamond C restoring

coherence Also the observer at finite time has not had an opportunity to observe

degrees of freedom coming from the portion outside his past lightcone on B but those

he might observe by waiting longer Hence we will be interested only in degrees of

freedom that leave C by crossing the boundary B

The boundaryB may contain components that are the event horizons of black holes

If black hole evaporation is unitary then such degrees of freedom will be returned to

the interior of the causal diamond in the form of Hawking radiation We can treat

this formally by replacing the black hole with a membrane that contains the relevant

degrees of freedom at the stretched horizon and releases them as it shrinks to zero

size [20] However we insist that degrees of freedom crossing the outermost component

of B (which corresponds to the event horizon in de Sitter universes) are traced over It

does not matter for this purpose whether we regard these degrees of freedom as being

absorbed by the boundary or as crossing through the boundary as long as we assume

ndash 14 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

Figure 6 The surface β divides the future boundary of the causal diamond into two portions

Bplusmn Degrees of freedom that passed through Bminus are forever inaccessible from within the

diamond Tracing over them defines a density matrix at the time γ The pure states that

diagonalize this matrix can be represented as branches As more degrees of freedom leave the

causal diamond a branching tree is generated that represents all possible decoherent histories

within the diamond

that they are inaccessible to any experiment performed within C This assumption

seems reasonable since there is no compelling argument that the unitarity evaporation

of black holes should extend to cosmological event horizons Indeed it is unclear how

the statement of unitarity would be formulated in that context (A contrary viewpoint

which ascribes unitarity even to non-Killing horizons is explored in Ref [11])

The boundary B is a null hypersurface Consider a cross-section β of B ie a

spacelike two-dimensional surface that divides B into two portions the upper portion

B+ which contains the tip of the causal diamond and the lower portion Bminus We

may trace over degrees of freedom on Bminus this corresponds to the matter that has

left the causal diamond by the time β and hence has become inaccessible from within

the diamond Thus we obtain a density matrix ρ(β) on the portion B+ Assuming

unitary evolution of closed systems the same density matrix also determines the state

on any spacelike surface bounded by β and it determines the state on the portion of

the boundary of the past of β that lies within C γ Note that γ is a null hypersurface

In fact γ can be chosen to be a future lightcone from an event inside C (more precisely

the portion of that light-cone that lies within C) the intersection of γ with B then

ndash 15 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

defines β

A useful way of thinking about ρ(β) is as follows The boundary of the causal past

of β consists of two portions γ and Bminus The degrees of freedom that cross Bminus are

analogous to the environment in the usual discussion of decoherence except in that

they are inaccessible from within the causal diamond C not just in practice but in

principle The remaining degrees of freedom in the past of β cross through γ and thus

stay inside the causal diamond They are analogous to the system and apparatus which

are now in one of the states represented in the density matrix ρ(β) A measurement

is an interaction between degrees of freedom that later pass through γ and degrees of

freedom that later pass through Bminus The basis in which ρ(β) is diagonal consists of the

different pure states that could result from the outcome of measurements in the causal

past of β

We can now go further and consider foliations of the boundary B Each member of

the foliation is a two-dimensional spatial surface β dividing the boundary into portions

B+ and Bminus We can regard β as a time variable For example any a foliation of the

causal diamond C into three-dimensional spacelike hypersurfaces of equal time β will

induce a foliation of the boundary B into two-dimensional spacelike surfaces Another

example on which we will focus is shown in Fig 6 consider an arbitrary time-like

worldline that ends at the tip of the causal diamond Now construct the future light-

cone from every point on the worldline This will induce a foliation of B into slices β

It is convenient to identify β with the proper time along the worldline

The sequence of density matrices ρ(β1) ρ(β2) ρ(βn) describes a branching tree

in which any path from the root to one of the final twigs represents a particular history of

the entire causal diamond coarse-grained on the appropriate timescale These histories

are ldquominimally decoherentrdquo in the sense that the only degrees of freedom that are traced

over are those that cannot be accessed even in principle In practice an observer at the

time β may well already assign a definite outcome to an observation even though no

particles correlated with the apparatus have yet crossed Bminus(β) There is a negligible

but nonzero probability of recoherence until the first particles cross the boundary only

then is coherence irreversibly lost

Strictly speaking the above analysis should be expanded to allow for the different

gravitational backreaction of different branches The exact location of the boundary B

at the time β depends on what happens at later times (This suggests that ultimately

it may be more natural to construct the decoherent causal diamonds from the top

down starting in the future and ending in the past) Here we will be interested mainly

in the application to the eternally inflating multiverse4 where we can sidestep this

issue by choosing large enough timesteps In de Sitter vacua on timescales of order4However the above discussion has implications for any global geometry in which observers fall

ndash 16 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

tΛ sim |Λ|minus12 the apparent horizon which is locally defined approaches the event

horizon B at an exponential rate Mathematically the difference between the two

depends on the future evolution but it is exponentially small and thus is irrelevant

physically Vacua with negative cosmological constant crunch on the timescale tΛ [21]

and so will not be resolved in detail at this level of coarse-graining

We expect that the distinction between causal diamond bulk and its boundary

is precise only to order eminusA(β) where A is the area of the boundary at the time β

Because of entropy bounds [622ndash24] no observables in any finite spacetime region can

be defined to better accuracy than this A related limitation applies to the objective

notion of decoherence we have given and it will be inherited by the reconstruction of

global geometry we propose below This will play an imporant role in Sec 3 where

we will argue that the hat regions with Λ = 0 provide an exact counterpart to the

approximate observables and approximate decoherence described here

25 Global-local measure duality

In this section we will review the duality that relates the causal diamond to a global

time cutoff called light-cone time both define the same probabilities when they are

used as regulators for eternal inflation As originally derived the duality assumed the

standard global picture as a starting point a viewpoint we have criticized in Sec 22

Here we will take the opposite viewpoint the local picture is the starting point and

the duality suggests that a global spacetime can be reconstructed from the more funda-

mental structure of decoherent causal diamond histories Indeed light-cone time will

play a central role in the construction proposed in Sec 26

By restricting to a causal diamond we obtained a natural choice of environment

the degrees of freedom that exit from the diamond Tracing over this environment leads

to a branching tree of objective observer-independent decoherent historiesmdashprecisely

the kind of notion that was lacking in the global description In the causal diamond

bubbles of different vacua really do nucleate at specific times and places They decohere

when the bubble wall leaves the diamond

Consider a large landscape of vacua Starting say in a vacuum with very large

cosmological constant a typical diamond contains a sequence of bubble nucleations

(perhaps hundreds in some toy models [1425]) which ends in a vacuum with negative

cosmological constant (and thus a crunch) or with vanishing cosmological constant (a

out of causal contact at late times including crunching universes and black hole interiors Suppose all

observers were originally causally connected ie their past light-cones substantially overlap at early

times Then the different classes of decoherent histories that may be experienced by different observers

arise from differences in the amount the identity and the order of the degrees of freedom that the

observer must trace over

ndash 17 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

future boundary

0

Q

Q( )

Figure 7 In this diagram the standard global picture of eternal inflation is taken as a

starting point Geodesics (thin vertical lines) emanating from an initial surface Σ0 define

an ensemble of causal patches (the leftmost is shaded greylight) with a particular mix of

initial conditions In the limit of a dense family of geodesics the global spacetime is thus

deconstructed into overlapping causal patches The number of patches overlapping at Q is

determined by the number of geodesics entering the future light-cone of Q In the continuum

limit this becomes the volume ε(Q) from which the relevant geodesics originate which in turn

defines the light-cone time at Q This relation implies an exact duality between the causal

patch measure and the light-cone time cut-off for the purpose of regulating divergences and

computing probabilities in eternal inflation

supersymmetric open universe or ldquohatrdquo) Different paths through the landscape are

followed with probabilities determined by branching ratios Some of these paths will

pass through long-lived vacua with anomalously small cosmological constant such as

ours

The causal diamond has already received some attention in the context of the mul-

tiverse It was proposed [26] as a probability measure a method for cutting off infinities

and obtaining well-defined amplitudes Phenomenologically the causal diamond mea-

sure is among the most successful proposals extant [27ndash31] From the viewpoint of

economy it is attractive since it merely exploits a restriction that was already imposed

on us by black hole complementarity and uses it to solve another problem And con-

ceptually our contrasting analyses of decoherence in the global and causal diamond

viewpoints suggests that the causal diamond is the more fundamental of the two

This argument is independent of black hole complementarity though both point at

the same conclusion It is also independent of the context of eternal inflation However

if we assume that the universe is eternally inflating then it may be possible to merge

all possible causal diamond histories into a single global geometry

ndash 18 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

If we are content to take the standard global picture as a starting point then it

is straightforward to deconstruct it into overlapping causal diamonds or patches5 [13

32] (see Fig 7 taken from Ref [32]) Indeed a stronger statement is possible as

far as any prediction goes the causal diamond viewpoint is indistinguishable from a

particular time cutoff on the eternally inflating global spacetime An exact duality [32]

dictates that relative probabilities computed from the causal diamond agree exactly

with the probabilities computed from the light-cone time cutoff6 The duality picks

out particular initial conditions for the causal diamond it holds only if one starts in

the ldquodominant vacuumrdquo which is the de Sitter vacuum with the longest lifetime

The light-cone time of an event Q is defined [34] in terms of the volume ε(Q) of

the future light-cone of Q on the future conformal boundary of the global spacetime

see Fig 7

t(Q) equiv minus1

3log ε(Q) (22)

The volume ε in turn is defined as the proper volume occupied by those geodesics

orthogonal to an initial hypersurface Σ0 that eventually enter the future of Q (For an

alternative definition directly in terms of an intrinsic boundary metric see Ref [35])

We emphasize again that in these definitions the standard global picture is taken for

granted we disregard for now the objections of Sec 22

The light-cone time of an event tells us the factor by which that event is overcounted

in the overlapping ensemble of diamonds This follows from the simple fact that the

geodesics whose causal diamonds includes Q are precisely the ones that enter the causal

future of Q Consider a discretization of the family of geodesics orthogonal to Σ0 into

a set of geodesics at constant finite density as shown in Fig 7 The definition of

light-cone time ensures that the number of diamonds that contain a given event Q is

proportional to ε = exp[minus3t(Q)] Now we take the limit as the density of geodesics

on Σ0 tends to infinity In this limit the entire global spacetime becomes covered by

the causal diamonds spanned by the geodesics The relative overcounting of events at

two different light-cone times is still given by a factor exp(minus3∆t) (To show that this

implies the exact equivalence of the causal diamond and the light-conetime cutoff one

must also demonstrate that the rate at which events of any type I occur depends only

on t This is indeed the case if Σ0 is chosen sufficiently late ie if initial conditions

5A causal patch can be viewed as the upper half of a causal diamond In practice the difference is

negligible but strictly the global-local duality holds for the patch not the diamond Here we use it

merely to motivate the construction of the following subsection which does use diamonds6It is worth noting that the light-cone time cutoff was not constructed with this equivalence in

mind Motivated by an analogy with the UVIR relation of the AdSCFT correspondence [33] light-

cone time was formulated as a measure proposal [34] before the exact duality with the causal diamond

was discovered [32]

ndash 19 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

on the patches conform to the global attractor regime Since we are not primarily

interested in the measure problem here we will not review this aspect of the proof see

Ref [32] for details)

Given the above deconstruction of the global spacetime it is tempting to identify

the eternally inflating multiverse with the many worlds of quantum mechanics if the

latter could somehow be related to branches in the wavefunction of the causal diamonds

Without decoherence however there is neither a consistent global picture (as shown

in Sec 22) nor a sensible way of picking out a preferred basis that would associate

ldquomany worldsrdquo to the pure quantum state of a causal diamond (Sec 24)7

We have already shown that decoherence at the causal diamond boundary leads

to distinct causal diamond histories or ldquoworldsrdquo To recover the global multiverse and

demonstrate that it can be viewed as a representation of the many causal diamond

worlds one must show that it is possible to join together these histories consistently

into a single spacetime This task is nontrivial In the following section we offer a

solution in a very simple setting we leave generalizations to more realistic models to

future work Our construction will not be precisely the inverse of the deconstruction

shown in Fig 7 for example there will be no overlaps However it is closely related

in particular we will reconstruct the global spacetime in discrete steps of light-cone

time

26 Constructing a global multiverse from many causal diamond worlds

In this section we will sketch a construction in 1 + 1 dimensions by which a global

picture emerges in constant increments of light-cone time (Fig 8) For simplicity we

will work on a fixed de Sitter background metric

ds2

`2= minus(log 2 dt)2 + 22t dx2 (23)

where ` is an arbitrary length scale A future light-cone from an event at the time t

grows to comoving size ε = 21minust so t represents light-cone time up to a trivial shift

and rescaling t = 1minus log2 ε We take the spatial coordinate to be noncompact though

our construction would not change significantly if x was compactified by identifying

x sim= x+ n for some integer n

The fixed background assumption allows us to separate the geometric problemmdash

building the above global metric from individual causal diamondsmdashfrom the problem of

matching matter configurations across the seams We will be interested in constructing

a global spacetime of infinite four-volume in the future but not int the past Therefore

7In this aspect our viewpoint differs from [13]

ndash 20 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

1

2

3

x

t

3212minus12 0 1

0

A C ED

F G

Figure 8 Construction of a global spacetime from decoherent causal diamond histories

Green squares indicate the origins of causal diamonds that tile the global de Sitter space Red

dots indicate discrete time steps of order tΛ at which decoherence occurs within individual

causal diamonds For example a nonzero amplitude for bubble nucleation in the region

bounded by ACDEFG entangles the null segment A with C either both will contain a

bubble or neither will Similarly D is entangled with E The density matrix for CD is

diagonal in a basis consisting of pure states in which a bubble either forms or does not

form This eliminates superpositions of the false (white) and true (yellow) vacuum Initial

conditions for new causal diamonds at green squares are controlled by entanglements such as

that of A with C Further details are described in the text

we take the geodesic generating each diamond to be maximally extended towards the

future but finite towards the past This means that the lower tips do not lie on the

past conformal boundary of de Sitter space Note that all such semi-infinite diamonds

are isometric

The geometric problem is particularly simple in 1+1 dimensions because it is pos-

sible to tile the global geometry precisely with causal diamonds with no overlaps We

will demonstrate this by listing explicitly the locations of the causal diamond tiles

All diamonds in our construction are generated by geodesics that are comoving in the

metric of Eq (23) if the origin of the diamond is at (t x) then its tip is at (infin x)

Hence we will label diamonds by the location of their origins shown as green squares

ndash 21 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

in Fig 8 They are located at the events

(x t) = (m 0) (24)

(x t) =

(2m+ 1

2n n

) (25)

where n runs over the positive integers and m runs over all integers From Fig 8 it

is easy to see that these diamonds tile the global spacetime covering every point with

t ge 1 precisely once except on the edges We now turn to the task of matching the

quantum states of matter at these seams

We will assume that there exists a metastable vacuum F which decays with a rate

Γ per unit Hubble time and Hubble volume to a terminal vacuum T which does not

decay We will work to order Γ neglecting collisions between bubbles of the T vacuum

A number of further simplifications are made below

Our construction will be iterative We begin by considering the causal diamond

with origin at (x t) = (0 0) In the spirit of Sec 24 we follow the generating geodesic

(green vertical line) for a time ∆t = 1 (corresponding to a proper time of order tΛ) to

the event t = 1 x = 0 marked by a red circle in Fig 8 The future light-cone of this

event divides the boundary of the (0 0) diamond into two portions Bplusmn as in Fig 6

Bminus itself consists of two disconnected portions which we label A and E Together

with segments C and D of the future light-cone ACDE forms a Cauchy surface of

the diamond The evolution from the bottom boundary FG to the surface ACDE is

unitary For definiteness we will assume that the state on FG is the false vacuum

though other initial conditions can easily be considered

The pure state on ACDE can be thought of as a superposition of the false vac-

uum with bubbles of true vacuum that nucleate somewhere in the region delimited by

ACDEFG To keep things simple we imagine that decays can occur only at three

points at each with probability Γ sim Γ3 at the origin (0 0) at the midpoint of

the edge F (minus14 1

2) and at the midpoint of G (1

4 1

2) We assume moreover that

the true vacuum occupies the entire future light-cone of a nucleation point In this

approximation the pure state on ACDE takes the form

(1minus 3Γ)12 |F〉A|F〉C |F〉D|F〉E +Γ12|T 〉A|T 〉C |F〉D|F〉E+ Γ12 |T 〉A|T 〉C |T 〉D|T 〉E +Γ12|F〉A|F〉C |T 〉D|T 〉E (26)

where the last three terms correspond to the possible nucleation points from left to

right

From the point of view of an observer in the (0 0) diamond the Hilbert space

factor AE should be traced out This results in a density matrix on the slice CD

ndash 22 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

which can be regarded as an initial condition for the smaller diamond beginning at the

point (x t) = (0 1)

ρ(0 1) = (1minus 3Γ) |F〉C |F〉D D〈F| C〈F|+ Γ |F〉C |T 〉D D〈T | C〈F|+ Γ |T 〉C |F〉D D〈F| C〈T |+ Γ |T 〉C |T 〉D D〈T | C〈T | (27)

The density matrix can be regarded as an ensemble of four pure states |F〉C |F〉D with

probability (1minus 3Γ) and |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D each with probability Γ

The same construction can be applied to every ldquozeroth generationrdquo causal diamond

the diamonds with origin at (m 0) with m integer Since their number is infinite we

can realize the ensemble of Eq (27) precisely in the emerging global spacetime by

assigning appropriate initial conditions to the ldquofirst generation sub-diamondsrdquo (m 1)

The state |F〉C |T 〉D is assigned to a fraction 1 minus 3Γ of the (m 1) diamonds and

each of the states |F〉C |T 〉D |T 〉C |F〉D |T 〉C |T 〉D is assigned to a fraction Γ of (m 1)

diamonds8

So far we have merely carried out the process described in Fig 6 for one time step

in each of the (m 0) diamonds resulting in initial conditions for the subdiamonds that

start at the red circles at (m 1) In order to obtain a global description we must also

ldquofill in the gapsrdquo between the (m 1) diamonds by specifying initial conditions for the

ldquofirst generation new diamondsrdquo that start at the green squares at (m+ 12 1) But their

initial conditions are completely determined by the entangled pure state on ACDE

Eq (26) and the identical pure states on the analogous Cauchy surfaces of the other

(m 0) diamonds Because of entanglement the state on C is the same as on A If

C is in the true vacuum then so is A and if C is in the false vacuum then so is A

The edges D and E are similarly entangled Thus the assignment of definite initial

conditions to the (m 1) diamonds completely determines the initial conditions on the

(m + 12) diamonds We have thus generated initial conditions for all first-generation

diamonds (those with n = 1) Now we simply repeat the entire procedure to obtain

initial conditions for the second generation (n = 2) and so on9

8We defer to future work the interesting question of whether further constraints should be imposed

on the statistics of this distribution For example for rational values of Γ the assignment of pure

states to diamonds could be made in a periodic fashion or at random subject only to the above

constraint on relative fractions9Note that this construction determines the initial conditions for all but a finite number of diamonds

In a more general setting it would select initial conditions in the dominant vacuum We thank Ben

Freivogel and I-Sheng Yang for pointing out a closely related observation

ndash 23 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

This will generate a fractal distribution of true vacuum bubbles of the type that is

usually associated with a global description (Fig 8) The manner in which this picture

arises is totally distinct from a naive unitary evolution of global time slices in which a

full superposition of false and true vacuum would persist (with time-dependent ampli-

tudes) The standard global picture can only be obtained by exploiting the decoherence

of causal diamonds while proliferating their number The multiverse is a patchwork of

infinitely many branching histories the many worlds of causal diamonds

The construction we have given is only a toy model The causal diamonds in higher

dimensions do not fit together neatly to fill the space-time as they do in 1+1 dimensions

so overlaps will have to be taken into account10 Moreover we have not considered

the backreaction on the gravitational field Finally in general the future boundary

is not everywhere spacelike but contains hats corresponding to supersymmetric vacua

with Λ = 0 Our view will be that the hat regions play a very special role that is

complementary both in the colloquial and in the technical sense to the construction

we have given here Any construction involving finite causal diamonds in necessarily

approximate We now turn to the potentially precise world of the Census Taker a

fictitious observer living in a hat region

3 The many worlds of the census taker

31 Decoherence and recoherence

In Sec 21 we noted that decoherence is subjective to the extent that the choice of

environment is based merely on the practical considerations of an actual observer We

then argued in Sec 24 that the boundary of a causal patch can be regarded as a

preferred environment that leads to a more objective form of decoherence However

there is another element of subjectivity in decoherence which we have not yet addressed

decoherence is reversible Whether and how soon coherence is restored depends on

the dynamical evolution governing the system and environment

Consider an optical interference experiment (shown in Fig 9) in which a light beam

reflects off two mirrors m1 and m2 and then illuminates a screen S There is no sense

in which one can say that a given photon takes one or the other routes

On the other hand if an observer or simple detector D interacts with the photon

and records which mirror the photon bounced off the interference is destroyed One

of the two possibilities is made real by the interaction and thereafter the observer may

10Although not in the context of the multiverse Banks and Fischler have suggested that an inter-

locking collection of causal diamonds with finite dimensional Hilbert spaces can be assembled into the

global structure of de Sitter space [36]

ndash 24 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

Figure 9 Three optical interference experiments mrsquos denote mirrors S is a screen and D

means detector In the first setup the coherence of the two paths is maintained In the second

setup a detector is placed at one mirror If the detector is treated as an environment and its

Hilbert space is traced over then the coherence of the superposition of photon paths is lost

and the interference pattern disappears In the third setup the detector reverts to its original

state when the photon passes it a second time During the (arbitrarily long) time when the

photon travels between m1D and m3 tracing over the detector leads to decoherence But

after the second pass through m1D coherence is restored so the screen will show the same

pattern as in the first setup

ignore the branch of the wave function which does not agree with the observation

Moreover if a second observer describes the whole experiment ( including the first

observer) as a single quantum system the second observerrsquos observation will entangle

it in a consistent way

Now letrsquos consider an unusual version of the experiment in which the upper arm

of the interferometer is replaced by a mirror-detector m1D which detects a photon and

deflects it toward mirror m3 From m3 the photon is reflected back to the detector

and then to the screen The detector is constructed so that if a single photon passes

through it it flips to a new state (it detects) but the next time a photon passes through

it flips back to the original state The detector is well-insulated from any environment

The lower arm of the interferometer also has the path length increased but without a

detector

ndash 25 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

Since if the photon goes through the detector it goes through twice at the end of

the experiment the detector is left in the original state of no-detection It is obvious

that in this case the interference is restored But there is something unusual going

on During an intermediate time interval the photon was entangled with the detector

The event of passing through the upper arm has been recorded and the photonrsquos wave

function has collapsed to an incoherent superposition But eventually the photon and

the detector are disentangled What happened was made to unhappen

This illustrates that in order to give objective meaning to an event such as the

detection of a photon it is not enough that the system becomes entangled with the

environment the system must become irreversibly entangled with the environment or

more precisely with some environment

32 Failure to irreversibly decohere A limitation of finite systems

The above example may seem contrived since it relied on the perfect isolation of

the detector from any larger environment and on the mirror m3 that ensured that

the detected photon cannot escape It would be impractical to arrange in a similar

manner for the recoherence of macroscopic superpositions since an enormous number

of particles would have to be carefully controlled by mirrors However if we are willing

to wait then recoherence is actually inevitable in any system that is dynamically closed

ie a system with finite maximum entropy at all times

For example consider a world inside a finite box with finite energy and perfectly

reflecting walls If the box is big enough and is initially out of thermal equilibrium

then during the return to equilibrium structures can form including galaxies planets

and observers Entanglements can form between subsystems but it is not hard to

see that they cannot be irreversible Such closed systems with finite energy have a

finite maximum entropy and for that reason the state vector will undergo quantum

recurrences Whatever state the system finds itself in after a suitably long time it

will return to the same state or to an arbitrarily close state The recurrence time is

bounded by exp(N) = exp(eSmax) where N is the dimension of the Hilbert space that

is explored and Smax is the maximum entropy

This has an important implication for the causal diamonds of Sec 24 We argued

that the diamond bulk decoheres when degrees of freedom cross the boundary B of

the diamond But consider a potential that just contains a single stable de Sitter

vacuum with cosmological constant Λ Then the maximum area of the boundary of

the diamond is the horizon area of empty de Sitter space and the maximum entropy

is Smax = Amax4 = 3πΛ This is the maximum total entropy [7] which is the sum of

the matter entropy in the bulk and the Bekenstein-Hawking entropy of the boundary

Assuming unitarity and ergodicity [37] this system is dynamically closed and periodic

ndash 26 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

recurrences are inevitable (See Ref [38ndash40] for a discussion of precision in de Sitter

space)

Next consider a landscape that contains vacua with positive and negative cosmo-

logical constant We assume for now that there are no vacua with Λ = 0 Then the

maximum area Amax of the causal diamond boundary B is given by the greater of Λminus1+

and Λminus2minus where Λ+ is the smallest positive value of Λ among all landscape vacua and

Λminus is the largest negative value [41] B is a null hypersurface with two-dimensional

spatial cross-sections and it can be thought of as the union of two light-sheets that

emanate from the cross-section of maximum area Therefore the entropy that passes

through B is bounded by Amax2 and hence is finite

What does finite entropy imply for the decoherence mechanism of Sec 24 If the

causal diamond were a unitary ergodic quantum system then it would again follow

that recurrences including the restoration of coherence are inevitable This is plausible

for causal diamonds that remain forever in inflating vacua but such diamonds form

a set of measure zero Generically causal diamonds will end up in terminal vacua

with negative cosmological constant hitting a big crunch singularity after finite time

In this case it is not clear that they admit a unitary quantum mechanical description

over arbitrarily long timescales so recurrences are not mandatory However since

ergodicity cannot be assumed in this case it seems plausible to us that there exists a

tiny non-zero probability for recurrences We expect that in each metastable de Sitter

vacuum the probability for re-coherence is given by the ratio of the decay timescale to

the recurrence timescale Typically this ratio is super-exponentially small [42] but it

does not vanish In this sense the objective decoherence of causal diamond histories

described in Sec 24 is not completely sharp

33 Sagredorsquos postulates

The next morning Simplicio and Salviati visit Sagredo to continue their discussion

Simplicio I have been pondering the idea we came up with yesterday and I am

convinced that we have completely solved the problem Causal diamonds have definite

histories obtained by tracing over their boundary which we treat as an observer-

independent environment This gets rid of superpositions of different macroscopic ob-

jects such as bubbles of different vacua without the need to appeal to actual observers

inside the diamond Each causal diamond history corresponds to a sequence of things

that ldquohappenrdquo And the global picture of the multiverse is just a representation of

all the possible diamond histories in a single geometry the many worlds of causal

diamonds

Sagredo I wish I could share in your satisfaction but I am uncomfortable Let

me describe my concerns and perhaps you will be able to address them

ndash 27 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

Salviati I too have been unable to shake off a sense that that this is not

the whole storymdashthat we should do better I would be most interested to hear your

thoughts Sagredo

Sagredo Itrsquos true as Simplicio says that things ldquohappenrdquo when we trace over

degrees of freedom that leave the causal diamond Pure states become a density matrix

or to put it in Bohrrsquos language the wavefunction describing the interior of the diamond

collapses But how do we know that the coherence will not be restored What prevents

things from ldquounhappeningrdquo later on

Simplicio According to Bohr the irreversible character of observation is due to

the large classical nature of the apparatus

Salviati And decoherence allows us to understand this rather vague statement

more precisely the apparatus becomes entangled with an enormous environment which

is infinite for all practical purposes

Sagredo But even a large apparatus is a quantum system and in principle

the entanglement can be undone The irreversibility of decoherence is often conflated

with the irreversibility of thermodynamics A large system of many degrees of freedom

is very unlikely to find its way back to a re-cohered state However thermodynamic

irreversibility is an idealization that is only true for infinite systems The irreversibility

of decoherence too is an approximation that becomes exact only for infinite systems

Simplicio But think how exquisite the approximation is In a causal diamond

containing our own history the boundary area becomes as large as billions of light

years squared or 10123 in fundamental units As you know I have studied all the

ancients I learned that the maximum area along the boundary of a past light-cone

provides a bound on the size N of the Hilbert space describing everything within the

light-cone N sim exp(10123) [7] And elsewhere I found that re-coherence occurs on a

timescale NN sim exp[exp(10123)] [2] This is much longer than the time it will take for

our vacuum to decay [1543] and the world to end in a crunch So why worry about it

Sagredo Itrsquos true that re-coherence is overwhelmingly unlikely in a causal di-

amond as large as ours But nothing you said convinces me that the probability for

things to ldquounhappenrdquo is exactly zero

Salviati To me it is very important to be able to say that some things really

do happen irreversibly and without any uncertainty If this were not true then how

could we ever make sense of predictions of a physical theory If we cannot be sure that

something happened how can we ever test the prediction that it should or should not

happen

Sagredo Thatrsquos itmdashthis is what bothered me The notion that things really

happen should be a fundamental principle and the implementation of fundamental

ndash 28 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

principles in a physical theory should not rely solely on approximations So let me

define this more carefully in terms of a definition and a postulate

Definition I Consider an instance of decoherence (or ldquocollapse of a wave functionrdquo)

in a Hilbert space HS which occurs as a result of entanglement with another Hilbert

space HE The event will be said to happen if the entanglement between HE and HS is

irreversible and the system S can then be treated as if it was in one of the pure states

that constitute the basis that diagonalizes the density matrix obtained by tracing over

the Hilbert space HE

Postulate I Things happen

In other words there exist some entanglements in Nature that will not be reversed with

any finite probability

Simplicio Let me see if I can find an example that satisfies your postulate

Suppose that an apparatus is in continuous interaction with some environment Even

an interaction with an single environmental photon can record the event If the photon

disappears to infinity so that no mirror can ever reflect it back then the event has

happened

Sagredo Your example makes it sound like irreversible decoherence is easy but

I donrsquot think this is true For example in the finite causal diamonds we considered

there is no ldquoinfinityrdquo and so nothing can get to it

Salviati Sagredorsquos postulate is in fact surprisingly strong Any dynamically

closed system (a system with finite entropy) cannot satisfy the postulate because

recurrences are inevitable This not a trivial point since it immediately rules out

certain cosmologies Stable de Sitter space is a closed system More generally if we

consider a landscape with only positive energy local minima the recurrence time is

controlled by the minimum with the smallest cosmological constant So the recurrence

time is finite and nothing happens Anti de Sitter space is no better As is well known

global AdS is a box with reflecting walls At any finite energy it has finite maximum

entropy and also gives rise to recurrences

Simplicio I can think of a cosmology that satisfies Postulate I along the lines

of my previous example I will call it ldquoS-matrix cosmologyrdquo It takes place in an

asymptotically flat spacetime and can described as a scattering event The initial

state is a large number of incoming stable particles The particles could be atoms of

hydrogen oxygen carbon etc The atoms come together and form a gas cloud that

contracts due to gravity Eventually it forms a solar system that may have observers

doing experiments Photons scatter or are emitted from the apparatuses and become

ndash 29 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

irreversibly entangled as they propagate to infinity in the final state The central star

collapses to a black hole and evaporates into Hawking radiation Eventually everything

becomes outgoing stable particles

Sagredo It is true that there are some things that happen in your S-matrix

cosmology But I am not satisfied I think there is a larger point that Salviati made

we would like a cosmology in which it is possible to give a precise operational meaning

to quantum mechanical predictions The notion that things unambiguously happen is

necessary for this but I now realize that it is not sufficient

Salviati Now Irsquom afraid you have us puzzled What is wrong with the S-matrix

cosmology

Sagredo Quantum mechanics makes probabilistic predictions when something

happens each possible outcome has probability given by the corresponding diagonal

entry in the density matrix But how do we verify that this outcome really happens

with predicted probability

Salviati Probabilities are frequencies they can be measured only by performing

an experiment many times For example to test the assertion that the probability

for ldquoheadsrdquo is one half you flip a coin a large number of times and see if within the

margin of error it comes up heads half of the time If for some reason it were only

possible to flip a coin once there would be no way to test the assertion reliably And to

be completely certain of the probability distribution it would be necessary to flip the

coin infinitely many times

Simplicio Well my S-matrix cosmology can be quite large For example it might

contain a planet on which someone flips a quantum coin a trillion times Photons record

this information and travel to infinity A trillion outcomes happen and you can do your

statistics Are you happy now

Sagredo A trillion is a good enough approximation to infinity for all practical

purposes But as I said before the operational testability of quantum-mechanical pre-

dictions should be a fundamental principle And the implementation of a fundamental

principle should not depend on approximations

Salviati I agree No matter how large and long-lived Simplicio makes his S-

matrix cosmology there will only be a finite number of coin flips And the cosmology

contains many ldquolargerrdquo experiments that are repeated even fewer times like the explo-

sion of stars So the situation is not much better than in real observational cosmology

For example inflation tells us that the quadrupole anisotropy of the CMB has a gaus-

sian probability distribution with a variance of a few times 10minus5 But it can only

be measured once so we are very far from being able to confirm this prediction with

complete precision

ndash 30 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

Sagredo Let me try to state this more precisely Quantum mechanics makes

probabilistic predictions which have the following operational definition

Definition II Let P (i) be the theoretical probability that outcome i happens (ie i

arises as a result of irreversible decoherence) given by a diagonal entry in the density

matrix Let N be the number of times the corresponding experiment is repeated and

let Ni be the number of times the outcome i happens The sharp prediction of quantum

mechanics is that

P (i) = limNrarrinfin

Ni

N (31)

Salviati Do you see the problem now Simplicio What bothers us about your

S-matrix cosmology is that N is finite for any experiment so it is impossible to verify

quantum-mechanical predictions with arbitrary precision

Simplicio Why not proliferate the S-matrix cosmology Instead of just one

asymptotically flat spacetime I will give you infinitely many replicas with identical

in-states They are completely disconnected from one another their only purpose is to

allow you to take N rarrinfin

Salviati Before we get to the problems let me say that there is one thing I like

about this proposal it provides a well-defined setting in which the many worlds inter-

pretation of quantum mechanics appears naturally For example suppose we measure

the z-component of the spin of an electron I have sometimes heard it said that when

decoherence occurs the world into two equally real branches But there are problems

with taking this viewpoint literally For example one might conclude that the proba-

bilities for the two branches are necessarily equal when we know that in general they

are not

Simplicio Yes I have always found this confusing So how does my proposal

help

Salviati The point is that in your setup there are an infinite number of worlds

to start with all with the same initial conditions Each world within this collection

does not split the collection S itself splits into two subsets In a fraction pi of worlds

the outcome i happens when a spin is measured There is no reason to add another

layer of replication

Simplicio Are you saying that you reject the many worlds interpretation

Salviati That depends on what you mean by it Some say that the many worlds

interpretation is a theory in which reality is the many-branched wavefunction itself I

dislike this idea because quantum mechanics is about observables like position mo-

mentum or spin The wavefunction is merely an auxiliary quantity that tells you

how to compute the probability for an actual observation The wavefunction itself is

ndash 31 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

not a measurable thing For example the wavefunction ψ(x) of a particle cannot be

measured so in particular one cannot measure that it has split But suppose we had a

system composed of an infinite number of particles all prepared in an identical manner

Then the single particle wavefunction ψ(x) becomes an observable for the larger sys-

tem For example the single particle probability density ψlowast(x)ψ(x) is a many-particle

observable

ψlowast(x)ψ(x) = limNrarrinfin

1

N

Nsumi=1

δ(xi minus x) (32)

Simplicio I see Now if an identical measurement is performed on each particle

each single particle wavefunction splits and this split wavefunction can be measured

in the many-particle system

Salviati To make this explicit we could make the individual systems more com-

plicated by adding a detector d for each particle Each particle-detector system can be

started in the product state ψ0(x)χ0(d) Allowing the particle to interact with the detec-

tor would create entanglement ie a wavefunction of the form ψ1(x)χ1(d)+ψ2(x)χ2(d)

But the branched wave function cannot be measured any better than the original one

Now consider an unbounded number of particle-detector pairs all starting in the same

product state and all winding up in entangled states It is easy to construct operators

analogous to Eq (32) in the product system that correspond to the single systemrsquos

wave function So you see that in the improved S-matrix cosmology there is no reason

to add another layer of many-worlds

Sagredo Thatrsquos all very nice but Simpliciorsquos proposal does not help with mak-

ing quantum mechanical predictions operationally well-defined You talk about many

disconnected worlds so by definition it is impossible to collect the N rarrinfin results and

compare their statistics to the theoretical prediction

Simplicio I see By saying that quantum mechanical predictions should be op-

erationally meaningful you mean not only that infinitely many outcomes happen but

that they are all accessible to an observer in a single universe

Sagredo Yes it seems to me that this requirement follows from Salviatirsquos fun-

damental principle that predictions should have precise operational meaning Let me

enshrine this in another postulate

Postulate II Observables are observable

By observables I mean any Hermitian operators whose probability distribution is pre-

cisely predicted by a quantum mechanical theory from given initial conditions And by

observable I mean that the world is big enough that the observable can be measured

infinitely many times by irreversible entanglement

ndash 32 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

Salviati Like your first postulate this one seems quite restrictive It obviously

rules out the infinite set of S-matrix cosmologies since products of field operators in

different replicas are predicted but cannot be measured by any observers in any one

cosmology And it gives us additional reasons to reject Simpliciorsquos earlier suggestion

the single S-matrix cosmology contains observables such as the total number of outgoing

particles which cannot even happen since there is no environment to measure them

Simplicio Well we were quite happy yesterday with the progress we had made

on making decoherence objective in the causal diamond But these postulates of yours

clearly cannot be rigorously satisfied in any causal diamond no matter how large

Perhaps itrsquos time to compromise Are you really sure that fundamental principles have

to have a completely sharp implementation in physical theory What if your postulates

simply cannot be satisfied

Salviati Yesterday we did not pay much attention to causal diamonds that end

in regions with vanishing cosmological constant Perhaps this was a mistake In such

regions the boundary area and entropy are both infinite I may not have not thought

about it hard enough but I see no reason why our postulates could not be satisfied in

these ldquohatrdquo regions

Sagredo Even if they can there would still be the question of whether this helps

make sense of the finite causal diamonds I care about this because I think we live in

one of them

Salviati I understand that but let us begin by asking whether our postulates

might be satisfied in the hat

34 Irreversible decoherence and infinite repetition in the hat

In the eternally inflating multiverse there are three types of causal diamonds The first

constitute a set of measure zero and remain forever in inflating regions The entropy

bound for such diamonds is the same as for the lowest-Λ de Sitter space they access

and hence is finite The second type are the diamonds who end on singular crunches If

the crunches originated from the decay of a de Sitter vacuum then the entropy bound

is again finite [44] Finally there are causal diamonds that end up in a supersymmetric

bubble of zero cosmological constant A timelike geodesic that enters such a bubble

will remain within the diamond for an infinite time It is convenient to associate an

observer with one of these geodesics called the Census Taker

The Census Takerrsquos past light-cone becomes arbitrarily large at late times It

asymptotes to the null portion of future infinity called the ldquohatrdquo and from there extends

down into the non-supersymmetric bulk of the multiverse (Fig 10) If the CTrsquos bubble

nucleates very late the entire hat appears very small on the conformal diagram and

the CTrsquos past light-cone looks similar to the past light-cone of a point on the future

ndash 33 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

Figure 10 The shaded region is the late time portion of a causal diamond that ends in a

supersymmetric vacuum with vanishing cosmological constant (left) We will think of this

region as the asymptotic causal past of an ldquoobserverrdquo in this vacuum called the Census Taker

A portion of the boundary of such causal diamonds has infinite cross-sectional area This

portion called a ldquohatrdquo coincides with a light-like portion of the future conformal boundary

in the standard global picture of the multiverse Λ = 0 bubbles that form late look small on

the conformal diagram (right) But they always have infinite area Hence the hat regions can

admit exact observables and can allow things to happen with enough room for arbitrarily

good statistics

boundary of ordinary de Sitter space However this is misleading the entropy bound

for the hat geometry is not the de Sitter entropy of the ancestor de Sitter space It is

infinite

According to the criterion of Ref [44] the existence of light-sheets with unbounded

area implies the existence of precise observables A holographic dual should exist

by which these observables are defined The conjecture of Ref [45]mdashthe FRWCFT

dualitymdashexhibits close similarities to AdSCFT In particular the spatial hypersur-

faces of the FRW geometry under the hat are open and thus have the form of three-

dimensional Euclidean anti-de Sitter space Their shared conformal boundarymdashthe

space-like infinity of the FRW geometrymdashis the two-dimensional ldquorimrdquo of the hat

As time progresses the past light-cone of the Census Taker intersects any fixed earlier

time slice closer and closer to the boundary For this reason the evolution of the Census

ndash 34 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

Takerrsquos observations has the form of a renormalization group flow in a two-dimensional

Euclidean conformal field theory [46]

We will return to the holographic dual later but for now we will simply adopt the

conjecture that exact versions of observables exist in the hat We can then ask whether

hat observables are observable in the sense of Postulate II above In particular this

would require that things happen in the hat in the sense of Postulate I We do not

understand the fundamental description of the hat well enough to prove anything rigor-

ously but we will give some arguments that make it plausible that both postulates can

indeed be satisfied Our arguments will be based on conventional (bulk) observables

Consider an interaction at some event X in the hat region that entangles an ap-

paratus with a photon If the photon gets out to the null conformal boundary then

a measurement has happened at X and postulate I is satisfied In general the photon

will interact with other matter but unless this interaction takes the form of carefully

arranged mirrors it will not lead to recoherence Instead will enlarge the environment

and sooner or later a massless particle that is entangled with the event at X will reach

the null boundary For postulate I to be satisfied it suffices that there exist some events

for which this is the case this seems like a rather weak assumption

By the FRW symmetry [21] of the hat geometry the same measurement happens

with the same initial conditions at an infinite number of other events Xi Moreover

any event in the hat region eventually enters the Census Takerrsquos past light-cone Let

N(t) be the number of equivalent measurements that are in the Census Takerrsquos past

light-cone at the time t along his worldline and let Ni(t) be the number of times the

outcome i happens in the same region Then the limit in Eq (31) can be realized as

pi = limtrarrinfin

Ni(t)

N(t) (33)

and Postulate II is satisfied for the particular observable measured

A crucial difference to the S-matrix cosmology discussed in the previous subsection

is that the above argument applies to any observable if it happens once it will happen

infinitely many times It does not matter how ldquolargerdquo the systems are that participate

in the interaction Because the total number of particles in the hat is infinite there

is no analogue of observables such as the total number of outgoing particles which

would be predicted by the theory but could not be measured This holds as long as

the fundamental theory predicts directly only observables in the hat which we assume

Thus we conclude that Postulate II is satisfied for all observables in the hat

Since both postulates are satisfied quantum mechanical predictions can be op-

erationally verified by the Census Taker to infinite precision But how do the hat

observables relate to the approximate observables in causal diamonds that do not enter

ndash 35 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

Figure 11 A black hole formed by gravitational collapse The horizonal shaded region is

the causal diamond of an observer (red) who remains forever outside the black hole The

vertical shaded region is the causal diamond of an observer (blue) who falls into the black

hole Note that there are infinitely many inequivalent infalling observers whose diamonds

have different endpoints on the conformal boundary (the singularity inside the black hole)

on the other hand all outside observers have the same causal diamond

hats and thus to the constructions of Sec 2 We will now argue that the non-hat

observables are approximations that have exact counterparts in the hat

35 Black hole complementarity and hat complementarity

In this subsection we will propose a complementarity principle that relates exact Cen-

sus Taker observables defined in the hat to the approximate observables that can be

defined in other types of causal diamonds which end in a crunch and have finite max-

imal area To motivate this proposal we will first discuss black hole complementarity

[20] in some detail

Can an observer outside the horizon of a black hole recover information about the

interior Consider a black hole that forms by the gravitational collapse of a star in

asymptotically flat space Fig 11) shows the spacetime region that can be probed by

an infalling observer and the region accessible to an observer who remains outside the

ndash 36 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

Figure 12 An observable A inside the black hole can be approximately defined and evolved

using local field theory In this way it can be propagated back out of the black hole to an

operator Ain defined on the asymptotic past boundary From there it is related by an exact

S-matrix to an operator Aout that can be measured in the Hawking radiation (wiggly dashed

line) by a late time observer

horizon At late times the two observers are out of causal contact but in the remote

past their causal diamonds have considerable overlap

Let A be an observable behind the horizon as shown in Fig 12 A might be a

slightly smeared field operator or product of such field operators To the freely falling

observer the observable A is a low energy operator that can be described by conventional

physics for example quantum electrodynamics

The question we want to ask is whether there is an operator outside the horizon

on future light-like infinity that has the same information as A Call it Aout By that

we mean an operator in the Hilbert space of the outgoing Hawking radiation that can

be measured by the outside observer and that has the same probability distribution

as the original operator A when measured by the in-falling observer

First we will show that there is an operator in the remote past Ain that has the

same probability distribution as A We work in the causal diamond of the infalling

observer in which all of the evolution leading to A is low energy physics Consider an

arbitrary foliation of the infalling causal diamond into Cauchy surfaces and let each

ndash 37 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

slice be labeled by a coordinate t We may choose t = 0 on the slice containing A

Let U(t) be the Schrodinger picture time-evolution operator and let |Ψ(t)〉 be the

state on the Cauchy surface t We can write |Ψ(0)〉 in terms of the state at a time minusTin the remote past

|Ψ(0)〉 = U(T )|Ψ(minusT )〉 (34)

The expectation value of A can be written in terms of this early-time state as

〈Ψ(minusT )|U dagger(T )AU(T )|Ψ(minusT )〉 (35)

Thus the operator

Ain = U dagger(T )AU(T ) (36)

has the same expectation value asA More generally the entire probability distributions

for A and Ain are the same Let us take the limit T rarr infin so that Ain becomes an

operator in the Hilbert space of incoming scattering states

Since the two diamonds overlap in the remote past Ain may also be thought of as

an operator in the space of states of the outside observer Now let us run the operator

forward in time by the same trick except working in the causal diamond of the outside

observer The connection between incoming and outgoing scattering states is through

the S-matrix Thus we define

Aout = SAinSdagger (37)

or

Aout = limTrarrinfin

SU dagger(T )AU(T )Sdagger (38)

The operator Aout when measured by an observer at asymptotically late time has the

same statistical properties as A if measured behind the horizon at time zero

The low energy time development operator U(T ) is relatively easy to compute since

it is determined by integrating the equations of motion of a conventional low energy

system such as QED However this part of the calculation will not be completely

precise because it involves states in the interior if the black hole which have finite

entropy bound The S-matrix should have a completely precise definition but is hard

to compute in practice Information that falls onto the horizon is radiated back out in

a completely scrambled form The black hole horizon is the most efficient scrambler of

any system in nature

This is the content of black hole complementarity observables behind the horizon

are not independent variables They are related to observables in front of the horizon by

unitary transformation The transformation matrix is limTrarrinfin[U(T )Sdagger] It is probably

not useful to say that measuring Aout tells us what happened behind the horizon [47]

It is not operationally possible to check whether a measurement of A and Aout agree

ndash 38 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

It is enough for us that the every (approximate) observable behind the horizon has a

(precise) complementary image among the degrees of freedom of the Hawking radiation

that preserves expectation values and probabilities

What is the most general form of operators A inside the black hole that can be

written in the form of Eq (38) as an operator Aout on the outside Naively we might

say any operator with support inside the black hole can be so represented since any

operator can be evolved back to the asymptotic past using local field theory But this

method is not completely exact and we know that there must be situations where

it breaks down completely For example by the same argument we would be free to

consider operators with support both in the Hawking radiation and in the collapsing

star and evolve them back this would lead us to conclude that either information

was xeroxed or lost to the outside This paradox is what led to the proposal that

only operators with support inside someonersquos causal patch make any sense But that

conclusion has to apply whether we are inside or outside the black hole the infalling

observer is not excepted For example we should not be allowed to consider operators at

large spacelike separation near the future singularity of the black hole The semiclassical

evolution back to the asymptotic past must be totally unreliable in this case11

We conclude that there are infinitely many inequivalent infalling observers with

different endpoints on the black hole singularity Approximate observables inside the

black hole must have the property that they can be represented by an operator with

support within the causal diamond of some infalling observer Any such operator can

be represented as an (exact) observable of the outside observer ie as an operator Aout

acting on the Hawking radiation on the outside

This completes our discussion of black hole complementarity Following Refs [46

48] we will now conjecture a similar relation for the multiverse The role of the ob-

server who remains outside the black hole will be played by the Census Taker note

that both have causal diamonds with infinite area The role of the many inequivalent

infalling observers will be played by the causal diamonds that end in crunches which

we considered in Sec 24 and which have finite area The conjecture is

Hat Complementarity Any (necessarily approximate) observable in the finite causal

diamonds of the multiverse can be represented by an exact observable in the Census

Takerrsquos hat

11The notion of causality itself may become approximate inside the black hole However this does

not give us licence to consider operators at large spacelike separation inside the black hole Large

black holes contain spatial regions that are at arbitrarily low curvature and are not contained inside

any single causal patch By the equivalence principle if we were permitted to violate the restriction

to causal patches inside a black hole then we would have to be allowed to violate it in any spacetime

ndash 39 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

1

3

2

t

4

Figure 13 Hat complementarity states that operators in finite causal diamonds (right)

have an exact counterpart in the hat Over time the census taker receives an unbounded

amount of information The past light-cones of the Census Taker at different times along his

worldline define a natural cutoff on his information Because there are only finitely many

different diamonds there must be infinitely many copies of every piece of information in the

hat They are related by hat complementarity to the infinite number of causal diamonds that

make up the global spacetime see also Fig 8 At late times increasing the Census Takerrsquos

cutoff is related to increasing the light-cone time cutoff which adds a new (redundant) layer

of diamonds

More precisely we assume that for every observable A in a finite causal diamond there

exists an operator Ahat with the same statistics as A A and Ahat are related the same

way that A and Aout are in the black hole case See Fig 13 for a schematic illustration

Hat complementarity is motivated by black hole complementarity but it does not

follow from it rigorously The settings differ in some respect For example a black

hole horizon is a quasi-static Killing horizon whereas the Census Takerrsquos causal hori-

zon rapidly grows to infinite area and then becomes part of the conformal boundary

Correspondingly the radiation in the hat need not be approximately thermal unlike

Hawking radiation And the argument that any operator behind the horizon can be

evolved back into the outside observerrsquos past has no obvious analogue for the Census

Taker since his horizon can have arbitrarily small area and hence contains very little

information at early times and since the global multiverse will generally contain re-

gions which are not in the causal future of the causal past of any Census Taker Here

we adopt hat complementarity as a conjecture

Next we turn to the question of how the information about finite causal diamonds

ndash 40 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

shows up in the precise description of the outside observer or the Census Taker

36 The global multiverse in a hat

An important difference between the black hole and the multiverse is that the infor-

mation in the Hawking radiation is finite whereas the number of particles in the hat

is infinite The observer outside the black hole therefore receives just enough infor-

mation to be able to recover the initial state and thus the approximate observables

inside the black hole On the other hand the O(3 1) symmetry of the FRW universe

implies that the entropy accessible to the Census Taker is infinite Since the number

of quantum states in the finite causal diamonds that end in crunches is bounded this

implies an infinite redundancy in the Census Takerrsquos information We will now argue

that this redundancy is related to the reconstruction of a global multiverse from causal

diamonds described in Sec 26 in which an infinite number of finite causal diamonds

are used to build the global geometry

Black holes scramble information Therefore an observer outside a black hole has

to wait until half the black hole has evaporated before the first bit of information can be

decoded in the Hawking radiation [20 49] After the whole black hole has evaporated

the outside observer has all the information and his information does not increase

further Can we make an analogous statement about the Census Taker

If the radiation visible to the Census Taker really is complementary to the causal

diamonds in the rest of the multiverse it will surely be in a similarly scrambled form

For hat complementarity to be operationally meaningful this information must be

organized in a ldquofriendlyrdquo way ie not maximally scrambled over an infinite-dimensional

Hilbert space Otherwise the census taker would have to wait infinitely long to extract

information about any causal patch outside his horizon12 This would be analogous to

the impossibility of extracting information from less than half of the Hawking radiation

An example of friendly packaging of information is not hard to come by (see also [50])

Imagine forever feeding a black hole with information at the same average rate that

it evaporates Since the entropy of the black hole is bounded it can never accumulate

more than S bits of information Any entering bit will be emitted in a finite time even

if the total number of emitted photons is infinite

12We are thus claiming that things happen in the past light-cone of the Census Taken at finite time

The corresponding irreversible entanglement leading to decoherence must be that between the interior

and the exterior of the Census Takerrsquos past light-cone on a suitable time-slice such as that shown in

Fig 13 Because the timeslice is an infinite FRW universe the environment is always infinite The

size of the ldquosystemrdquo on the other hand grows without bound at late times allowing for an infinite

number of decoherent events to take place in the Census Takerrsquos past Thus the postulates of Sec 33

can be satisfied

ndash 41 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

We will assume that this is also the case in the hat Then we can ask whether the

Census Takerrsquos cutoff translates to a multiverse cutoff hopefully something like the

light-cone time cutoff of Sec 25

To understand the Census Takerrsquos cutoff we start with the metric of open FRW

space with a vanishing cosmological constant We assume the FRW bubble nucleated

from some ancestor de Sitter vacuum

ds2 = a(T )2(minusdT 2 + dH23) (39)

where H3 is the unit hyperbolic geometry (Euclidean AdS)

dH23 = dR2 + sinh2R dΩ2

2 (310)

and T is conformal time The spatial hypersurfaces are homogeneous with symmetry

O(3 1) which acts on the two-dimensional boundary as special conformal transforma-

tions Matter in the hat fills the noncompact spatial slices uniformly and therefore

carries an infinite entropy

If there is no period of slow-roll inflation in the hat then the density of photons

will be about one per comoving volume We take the Census Taker to be a comoving

worldline (see Fig 13) The number of photons in the Census Takerrsquos causal past is

Nγ sim e2TCT (311)

In this formula TCT is the conformal time from which the Census Taker looks back Nγ

represents the maximum number of photons that the Census Taker can detect by the

time TCT

The Census Takerrsquos cutoff is an information cutoff after time TCT a diligent Census

Taker can have gathered about e2TCT bits of information If the de Sitter entropy of

the ancestor is Sa then after a conformal time TCT sim logSa the Census Taker will

have accessed an amount of information equal to the entropy of the causal patch of the

ancestor Any information gathered after that must be about causal diamonds in the

rest of the multiverse in the sense that it concerns operators like A that are beyond

the horizon

Over time the Census Taker receives an unbounded amount of information larger

than the entropy bound on any of the finite causal diamonds beyond the hat This

means that the Census Taker will receive information about each patch history over

and over again redundantly This is reminiscent of the fact that in our reconstruction

of the global picture every type of causal diamond history occurs over and over again

as the new diamonds are inserted in between the old ones This is obvious because we

ndash 42 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

used infinitely many diamonds to cover the global geometry and there are only finitely

many histories that end in a crunch or on an eternal endpoint

A particular history of a causal diamond has a larger or smaller maximum area

not a fixed amount of information But in our reconstruction of the global multiverse

each generation of new causal diamonds (the set of diamonds starting at the green

squares at equal time t in Fig 8) contains all possible histories (at least for sufficiently

late generations where the number of diamonds is large) Therefore the amount of

information in the reconstructed global geometry grows very simply like 2t This

redundant information should show up in the hat organized in the same manner Thus

it is natural to conjecture that the information cutoff of the Census Taker is dual by

complementarity to the discretized light-cone time cutoff implicit in our reconstruction

of a global spacetime from finite causal diamonds

Acknowledgments

We would like to thank T Banks B Freivogel A Guth D Harlow P Hayden

S Leichenauer V Rosenhaus S Shenker D Stanford E Witten and I Yang for

helpful discussions This work was supported by the Berkeley Center for Theoretical

Physics by the National Science Foundation (award numbers 0855653 and 0756174)

by fqxi grant RFP2-08-06 and by the US Department of Energy under Contract DE-

AC02-05CH11231

References

[1] L Susskind The Cosmic Landscape Chapter 11 Little Brown and Company New

York 2005

[2] W H Zurek ldquoDecoherence einselection and the quantum origins of the classicalrdquo

Rev Mod Phys 75 (2003) 715ndash775

[3] M Schlosshauer ldquoDecoherence the Measurement Problem and Interpretations of

Quantum Mechanicsrdquo Rev Mod Phys 76 (2004) 1267ndash1305 quant-ph0312059

[4] J Preskill ldquoLecture notes on quantum computationrdquo

httpwwwtheorycaltechedu~preskillph229lecture 1998

[5] W H Zurek ldquoPointer Basis of Quantum Apparatus Into What Mixture Does the

Wave Packet Collapserdquo Phys Rev D24 (1981) 1516ndash1525

[6] R Bousso ldquoHolography in general space-timesrdquo JHEP 06 (1999) 028

hep-th9906022

ndash 43 ndash

[7] R Bousso ldquoPositive vacuum energy and the N-boundrdquo JHEP 11 (2000) 038

hep-th0010252

[8] T Banks ldquoCosmological breaking of supersymmetry or little Lambda goes back to the

future IIrdquo hep-th0007146

[9] W Fischler ldquoTaking de Sitter seriously Talk given at Role of Scaling Laws in Physics

and Biology (Celebrating the 60th Birthday of Geoffrey West) Santa Fe Dec 2000rdquo

[10] T Banks ldquoCosmological breaking of supersymmetryrdquo Int J Mod Phys A16 (2001)

910ndash921

[11] L Susskind ldquoThe Census Takerrsquos Hatrdquo arXiv07101129 [hep-th]

[12] A Aguirre M Tegmark and D Layzer ldquoBorn in an Infinite Universe a Cosmological

Interpretation of Quantum Mechanicsrdquo arXiv10081066 [quant-ph]

[13] Y Nomura ldquoPhysical Theories Eternal Inflation and Quantum Universerdquo

arXiv11042324 [hep-th]

[14] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamical

neutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[15] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe Sitter vacua in string theoryrdquo

Phys Rev D 68 (2003) 046005 hep-th0301240

[16] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[17] F Denef and M R Douglas ldquoDistributions of flux vacuardquo JHEP 05 (2004) 072

hep-th0404116

[18] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow

first-order phase transitionrdquo Nucl Phys B212 (1983) 321ndash364

[19] W Wootters and W Zurek ldquoA single quantum cannot be clonedrdquo Nature 299 (1982)

802ndash803

[20] L Susskind L Thorlacius and J Uglum ldquoThe Stretched horizon and black hole

complementarityrdquo Phys Rev D 48 (1993) 3743 hep-th9306069

[21] S Coleman and F D Luccia ldquoGravitational effects on and of vacuum decayrdquo Phys

Rev D 21 (1980) 3305ndash3315

[22] G rsquot Hooft ldquoDimensional reduction in quantum gravityrdquo gr-qc9310026

[23] L Susskind ldquoThe World as a hologramrdquo J Math Phys 36 (1995) 6377ndash6396

hep-th9409089

ndash 44 ndash

[24] R Bousso ldquoA covariant entropy conjecturerdquo JHEP 07 (1999) 004 hep-th9905177

[25] R Bousso and I-S Yang ldquoLandscape Predictions from Cosmological Vacuum

Selectionrdquo Phys Rev D 75 (2007) 123520 hep-th0703206

[26] R Bousso ldquoHolographic probabilities in eternal inflationrdquo Phys Rev Lett 97 (2006)

191302 hep-th0605263

[27] R Bousso R Harnik G D Kribs and G Perez ldquoPredicting the cosmological constant

from the causal entropic principlerdquo Phys Rev D 76 (2007) 043513 hep-th0702115

[28] R Bousso and S Leichenauer ldquoPredictions from Star Formation in the Multiverserdquo

arXiv09074917 [hep-th]

[29] R Bousso L J Hall and Y Nomura ldquoMultiverse Understanding of Cosmological

Coincidencesrdquo Phys Rev D80 (2009) 063510 arXiv09022263 [hep-th]

[30] R Bousso and R Harnik ldquoThe Entropic Landscaperdquo Phys Rev D82 (2010) 123523

arXiv10011155 [hep-th]

[31] R Bousso B Freivogel S Leichenauer and V Rosenhaus ldquoA geometric solution to

the coincidence problem and the size of the landscape as the origin of hierarchyrdquo

Phys Rev Lett 106 (2011) 101301 arXiv10110714 [hep-th]

[32] R Bousso and I-S Yang ldquoGlobal-Local Duality in Eternal Inflationrdquo Phys Rev D80

(2009) 124024 arXiv09042386 [hep-th]

[33] J Garriga and A Vilenkin ldquoHolographic Multiverserdquo JCAP 0901 (2009) 021

arXiv08094257 [hep-th]

[34] R Bousso ldquoComplementarity in the Multiverserdquo Phys Rev D79 (2009) 123524

arXiv09014806 [hep-th]

[35] R Bousso B Freivogel S Leichenauer and V Rosenhaus ldquoBoundary definition of a

multiverse measurerdquo arXiv10052783 [hep-th]

[36] T Banks and W Fischler ldquoAn holographic cosmologyrdquo hep-th0111142

[37] L Dyson M Kleban and L Susskind ldquoDisturbing implications of a cosmological

constantrdquo JHEP 10 (2002) 011 hep-th0208013

[38] T Banks and W Fischler ldquoM-theory observables for cosmological space-timesrdquo

hep-th0102077

[39] T Banks W Fischler and S Paban ldquoRecurrent nightmares Measurement theory in

de Sitter spacerdquo JHEP 12 (2002) 062 arXivhep-th0210160

ndash 45 ndash

[40] T Banks W Fischler and L Mannelli ldquoMicroscopic quantum mechanics of the p =

rho universerdquo Phys Rev D71 (2005) 123514 arXivhep-th0408076

[41] R Bousso B Freivogel and S Leichenauer ldquoSaturating the holographic entropy

boundrdquo Phys Rev D82 (2010) 084024 arXiv10033012 [hep-th]

[42] R Bousso and B Freivogel ldquoA paradox in the global description of the multiverserdquo

JHEP 06 (2007) 018 hep-th0610132

[43] B Freivogel and M Lippert ldquoEvidence for a bound on the lifetime of de Sitter spacerdquo

JHEP 12 (2008) 096 arXiv08071104 [hep-th]

[44] D Harlow and L Susskind ldquoCrunches Hats and a Conjecturerdquo arXiv10125302

[hep-th]

[45] B Freivogel Y Sekino L Susskind and C-P Yeh ldquoA holographic framework for

eternal inflationrdquo Phys Rev D 74 (2006) 086003 hep-th0606204

[46] Y Sekino and L Susskind ldquoCensus Taking in the Hat FRWCFT Dualityrdquo Phys

Rev D80 (2009) 083531 arXiv09083844 [hep-th]

[47] R Bousso B Freivogel and I-S Yang ldquoEternal inflation The inside storyrdquo Phys

Rev D 74 (2006) 103516 hep-th0606114

[48] B Freivogel and L Susskind ldquoA framework for the landscaperdquo hep-th0408133

[49] D N Page ldquoExpected entropy of a subsystemrdquo Phys Rev Lett 71 (1993) 1291ndash1294

gr-qc9305007

[50] P Hayden and J Preskill ldquoBlack holes as mirrors quantum information in random

subsystemsrdquo JHEP 09 (2007) 120 arXiv07084025 [hep-th]

ndash 46 ndash